著名的 Lisp hacker – Zach Beane 为了教他十几岁的女儿学习 Lisp 编程，编写了一个入门教程，不过在我这个初学者眼里看来同样是值得一看的。

Lisp in Small Parts
20120612 10:36:00著名的 Lisp hacker – Zach Beane 为了教他十几岁的女儿学习 Lisp 编程，编写了一个入门教程，不过在我这个初学者眼里看来同样是值得一看的。 http://lisp.plasticki.com/... 
Small rectangular grey area covering parts of the side bar title
20201209 13:02:31<div><p><img alt="inbox1" src="https://imgblog.csdnimg.cn/img_convert/8143c8f8375aee6b8dc83b1abdc7d3dc.png" /></p>该提问来源于开源项目：boukestam/inboxingmail</p></div> 
Ubuntu Compress a big file into plenty of small parts（压缩文件为几个小部分,）
20150713 21:29:52由于百度网盘的上传文件大小限制，所以有些大文件不能愉快的上传，但是方法总比问题多，哈哈，最终找到了方法。 1.安装rar sudo aptget install rar ...2.右键点击要压缩的文件夹，选compress，然后右边后缀名...由于百度网盘的上传文件大小限制，所以有些大文件不能愉快的上传，但是方法总比问题多，哈哈，最终找到了方法。
1.安装rar
sudo aptget install rar
2.右键点击要压缩的文件夹，选compress，然后右边后缀名选择rar，最下面的split into volumes of ( ) MB输入你想分割的大小就好啦！

Low volume plastic aftermarket auto parts manufacture
20201024 08:19:49Vacuum Casting Technology is a fast molding method for producing small lot production. Mold Material is Silicone, it also called Silicone mold. Good option for discontinued auto parts producing. No ...Vacuum Casting Technology is a fast molding method for producing small lot production.
Mold Material is Silicone, it also called Silicone mold.
Good option for discontinued auto parts producing. No matter making sample upon customer samples or making very low volume product.Vacuum Casting Procedure Steps:
 Putting sample part into a container
 Add liquid silicone into container
 Putting them into the Vacuum furnace to get solid silicone Square.
 Separate silicone square piece to 2 pieces, top and bottom, as parting of injection mold
 Remove the sample part, then all part feature was left in the silicone pieces
Your Vacuum Casting silicone mold is ready.
 Preparing your assigned plastic liquid material, pour into the silicone mold
 In the vacuum furnace, liquid plastic filled the cavity in the silicone mold
 Once it was cooling, copied sample part was made
Usually, a silicone mold can make around 10+ pcs good parts, before dimension changed
•Advantage: Fast, lower cost on mold manufacture and small lot production
•Mold Raw Material : Liquid Silicone
•Production Material: All Plastic, especially for Rubber and Silicone parts.
•Usage:
Research & Development of new products (work with 3D printing sample)
Copy current product

MTMPartsLibrary源码
20210516 15:27:07SMALL_DIODE用于1N4148等信号二极管 用于VTL5C3和类似vactrol的VACTROL_4PIN也将适合Silonex NSL32SR3 VACTROL_5PIN适用于VTL5C3 / 2和类似的五针vactrol Radio Music的YAMAICHI PJS008U3000 
Small Ramsey Numbers
20060913 10:02:00Small Ramsey NumbersStanisław P. RadziszowskiDepartment of Computer ScienceRochester Institute of TechnologyRochester, NY 14623, spr@cs.rit.eduSubmitted: June 11, 1994; Accepted: July 3, 1994RevisionSmall Ramsey Numbers
Stanisław P. Radziszowski
Department of Computer Science
Rochester Institute of Technology
Rochester, NY 14623, spr@cs.rit.edu
Submitted: June 11, 1994; Accepted: July 3, 1994
Revision #10: July 4, 2004
ABSTRACT: We present data which, to the best of our knowledge,
includes all known nontrivial values and bounds for specific graph,
hypergraph and multicolor Ramsey numbers, where the avoided
graphs are complete or complete without one edge. Many results pertaining
to other more studied cases are also presented. We give references
to all cited bounds and values, as well as to previous similar
compilations. We do not attempt complete coverage of asymptotic
behavior of Ramsey numbers, but concentrate on their specific values.
Mathematical Reviews Subject Number 05C55.
Revisions
1993, February preliminary version, RITTR93009 [Ra2]
1994, July 3 accepted to the ElJC, posted on the web
1994, November 7 ElJC revision #1
1995, August 28 ElJC revision #2
1996, March 25 ElJC revision #3
1997, July 11 ElJC revision #4
1998, July 9 ElJC revision #5
1999, July 5 ElJC revision #6
2000, July 25 ElJC revision #7
2001, July 12 ElJC revision #8
2002, July 15 ElJC revision #9
2004, July 4 ElJC revision #10
 1 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
Contents
1. Scope and Notation 3
2. Classical Two Color Ramsey Numbers 4
2.1 Upper and lower bounds on R (k , l ) for k £ 10, l £ 15 4
2.2 Lower bounds on R (k , l ) for l ³ 15 6
2.3 Other results on R (k , l ) 7
3. Two Colors  Dropping One Edge from Complete Graph 8
4. General Graph Numbers in Two Colors 10
4.1 Paths 10
4.2 Cycles 10
4.3 Wheels 10
4.4 Books 11
4.5 Complete bipartite graphs 11
4.6 Triangle versus other graphs 12
4.7 Paths versus other graphs 12
4.8 Cycles versus complete graphs 13
4.9 Cycles versus other graphs 14
4.10 Stars versus other graphs 14
4.11 Books versus other graphs 15
4.12 Wheels versus other graphs 15
4.13 Trees and Forests 15
4.14 Mixed special cases 16
4.15 Mixed general cases 16
4.16 Other general results 17
5. Multicolor Graph Numbers 19
5.1 Bounds for multicolor classical numbers 19
5.2 General multicolor results for complete graphs 21
5.3 Special multicolor cases 22
5.4 General multicolor results for cycles and paths 22
5.5 Other general multicolor results 23
6. Hypergraph Numbers 24
7. Cumulative Data and Surveys 25
7.1 Cumulative data for two colors 25
7.2 Cumulative data for three colors 26
7.3 Surveys 27
8. Concluding Remarks 28
9. References 28
A through D 28
E through G 32
H through L 37
M through R 41
S through Z 44
 2 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
1. Scope and Notation
There is a vast literature on Ramsey type problems starting in 1930 with the original
paper of Ramsey [Ram]. Graham, Rothschild and Spencer in their book [GRS] present an
exciting development of Ramsey Theory. The subject has grown amazingly, in particular with
regard to asymptotic bounds for various types of Ramsey numbers (see the survey papers
[GrRo¨, Nes˘, ChGra2]), but the progress on evaluating the basic numbers themselves has been
very unsatisfactory for a long time. In the last two decades, however, considerable progress
has been obtained in this area, mostly by employing computer algorithms. The few known
exact values and several bounds for different numbers are scattered among many technical
papers. This compilation is a fast source of references for the best results known for specific
numbers. It is not supposed to serve as a source of definitions or theorems, but these can be
easily accessed via the references gathered here.
Ramsey Theory studies conditions when a combinatorial object contains necessarily some
smaller given objects. The role of Ramsey numbers is to quantify some of the general existential
theorems in Ramsey Theory.
Let G1,G2, . . . , Gm be graphs or s uniform hypergraphs (s is the number of vertices
in each edge). R (G1,G2, . . . , Gm ; s ) denotes the mcolor Ramsey number for s uniform
graphs/hypergraphs, avoiding Gi in color i for 1£ i £m. It is defined as the least integer n
such that, in any coloring with m colors of the s subsets of a set of n elements, for some i
the s subsets of color i contain a sub(hyper)graph isomorphic to Gi (not necessarily
induced). The value of R (G1,G2, . . . , Gm ; s ) is fixed under permutations of the first m
arguments.
If s = 2 (standard graphs) then s can be omitted. If Gi is a complete graph Kk , then we
can write k instead of Gi , and if Gi =G for all i we can use the abbreviation Rm (G; s ) or
Rm (G). For s = 2, Kk  e denotes a Kk without one edge, and for s = 3, Kk  t denotes a Kk
without one triangle (hyperedge). Pi is a path on i vertices, Ci is a cycle of length i , and Wi
is a wheel with i 1 spokes, i.e. a graph formed by some vertex x , connected to all vertices of
some cycle Ci 1. Kn ,m is a complete n by m bipartite graph, in particular K1,n is a star
graph. The book graph Bi = K2 + Ki = K1 + K1,i has i + 2 vertices, and can be seen as i triangular
pages attached to a single edge. The fan graph Fn is defined by Fn = K1 + nK2. For
a graph G, n (G) and e (G) denote the number of vertices and edges, respectively. Finally, let
c(G) be the chromatic number of G, and let nG denote n disjoint copies of G.
Section 2 contains the data for the classical two color Ramsey numbers R (k , l ) for complete
graphs, and section 3 for the two color case when the avoided graphs are complete or
have the form Kk  e , but not both are complete. Section 4 lists the most studied two color
cases for other graphs. The multicolor and hypergraph cases are gathered in sections 5 and 6,
respectively. Finally, section 7 gives pointers to cumulative data and to most of the previous
surveys.
 3 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
2. Classical Two Color Ramsey Numbers
2.1. Upper and lower bounds on R (k , l )
l 3 4 5 6 7 8 9 10 11 12 13 14 15
k
40 46 52 59 66 73
3 6 9 14 18 23 28 36
43 51 59 69 78 88
35 49 56 69 92 97 128 133 141 153
4 18 25
41 61 84 115 149 191 238 291 349 417
43 58 80 101 121 141 157 181 205 233 261
5
49 87 143 216 316 442
102 111 127 169 178 253 262 317 401
6
165 298 495 780 1171
205 216 232 405 416 511
7
540 1031 1713 2826
282 317 817 861
8
1870 3583 6090
565 580
9
6588 12677
798 1265
10
23556
Table I. Known nontrivial values and bounds for two color
Ramsey numbers R (k , l ) = R (k , l ; 2).
l 4 5 6 7 8 9 10 11 12 13 14 15
k
Ka2 GR Ka2 Ex5 Ka2 Ex12 Piw1 Ex8 WW
3 GG GG Ke´ry
GY MZ GR RK2 RK2 Les RK2 RK2 Les
Ka1 Ex9 Ex3 Ex15 RK1 HaKr 2.3.e SLL2 2.3.e XXR XXR
4 GG
MR4 MR5 Mac Mac Mac Mac Spe3 Spe3 Spe3 Spe3 Spe3
Ex4 Ex9 CET HaKr Haa Ex12 XXER Ex12 XXER XXER XXER
5
MR5 HZ1 Spe3 Spe3 Mac Mac
Ka1 2.3.e XXR XXER 2.3.e XXR 2.3.e XXER 2.3.h
6
Mac Mac Mac Mac Mac
She1 2.3.e 2.3.g XXER 2.3.e XXR
7
Mac Mac HZ1 Mac
BR XXER XXER 2.3.h
8
Mac Ea1 HZ1
She1 2.3.e
9
ShZ1 Ea1
She1 2.3.h
10
Shi2
References for Table I.
 4 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
We split the data into the table of values and a table with corresponding references. In
Table I, known exact values appear as centered entries, lower bounds as top entries, and upper
bounds as bottom entries.
The task of proving R (3, 3) £ 6 was the second problem in Part I of the William Lowell
Putnam Mathematical Competition held in March 1953 [Bush].
All the critical graphs for the numbers R (k , l ) (graphs on R (k , l )  1 vertices without Kk
and without Kl in the complement) are known for k = 3 and l = 3, 4, 5 [Ke´ry], 6 [Ka2], 7
[RK3, MZ], and there are 1, 3, 1, 7 and 191 of them, respectively. All (3, k )graphs, for
k £ 6, were enumerated in [RK3], and all (4,4)graphs in [MR2]. There exists a unique critical
graph for R (4,4) [Ka2]. There are 430215 such graphs known for R (3,8) [McK], 1 for
R (3,9) [Ka2] and 350904 for R (4, 5) [MR4], but there might be more of them. In [MR5] evidence
is given for the conjecture that R (5, 5) = 43 and that there exist 656 critical graphs on
42 vertices. The graphs constructed by Exoo in [Ex9, Ex12, Ex13, Ex14, Ex15, Ex16], and
some others, are available electronically from http://ginger.indstate.edu/ge/RAMSEY.
The construction by Mathon [Mat] and Shearer [She1] (see also sections 2.3.i, 5.2.h and
5.2.i), using data obtained by Shearer [She1], gives the following lower bounds for higher
diagonal numbers: R (11,11) ³ 1597, R (13,13) ³ 2557, R (14,14) ³ 2989, R (15,15) ³ 5485,
and R (16,16) ³ 5605. Similarly, R (17,17) ³ 8917, R (18,18) ³ 11005 and R (19,19) ³ 17885
were obtained in [LSL]. The same approach does not improve on an easy bound
R (12,12) ³ 1637 [XXR], which can be obtained by applying twice 2.3.e. Only some of the
higher bounds implied by 2.3.* are shown, and more similar bounds could be easily derived.
In general, we show bounds beyond the contiguous small values if they improve on results
previously reported in this survey or published elsewhere. Some easy upper bounds implied
by 2.3.a are marked as [Ea1].
Cyclic (or circular ) graphs are often used for Ramsey graph constructions. Several
cyclic graphs establishing lower bounds were given in the Ph.D. dissertation by J.G.
Kalbfleisch in 1966, and many others were published in the next few decades. Only recently
Harborth and Krause [HaKr] presented all best lower bounds up to 102 from cyclic graphs
avoiding complete graphs. In particular, no lower bound in Table I can be improved with a
cyclic graph on less than 102 vertices. See also item 2.3.k and section 4.16 [HaKr].
The claim that R (5, 5) = 50 posted on the web [Stone] is in error, and despite being
shown so more than once, this incorrect value is being cited by some authors. The bound
R (3, 13) ³ 60 [XZ] cited in the 1995 version of this survey was shown to be incorrect in
[Piw1]. Another incorrect construction for R (3, 10) ³ 41 was described in [DuHu].
There are really only two general upper bound inequalities useful for small parameters,
namely 2.3.a and 2.3.b. Stronger upper bounds for specific parameters were difficult to
obtain, and they often involved massive computations, like those for the cases of (3,8) [MZ],
(4,5) [MR4], (4,6) and (5,5) [MR5]. The bound R (6, 6) £ 166, only 1 more than the best
known [Mac], is an easy consequence of a theorem in [Walk] (2.3.b) and R (4, 6) £ 41. T.
Spencer [Spe3], Mackey [Mac], and Huang and Zhang [HZ1], using the bounds for minimum
and maximum number of edges in (4,5) Ramsey graphs listed in [MR3, MR5], were able to
establish new upper bounds for several higher Ramsey numbers, improving on all of the
 5 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
previous longstanding results by Giraud [Gi3, Gi5, Gi6].
We have recomputed the upper bounds in Table I marked [HZ1] using the method from
the paper [HZ1], because the bounds there relied on an overly optimistic personal communication
from T. Spencer. Further refinements of this method are studied in [HZ2, ShZ1, Shi2].
The paper [Shi2] subsumes the main results of the manuscripts [ShZ1, Shi2].
2.2. Lower bounds on R (k , l ), higher parameters
The lower bounds marked [XXR], [XXER], 2.3.e and 2.3.h need not to be cyclic, while
all other lower bounds listed in Table II were obtained by construction of cyclic graphs.
l 15 16 17 18 19 20 21 22 23
k
73 79 92 98 106 109 122 125 136
3
WW WW WWY1 WWY1 WWY1 WWY1 WWY1 WWY1 WWY1
153 182 187 198 230 242 282
4
XXR LSS 2.3.e LSZL SLZL SLZL SL
261 289 313 365 389 421 433 485 509
5
XXER 2.3.h 2.3.h 2.3.h 2.3.h 2.3.h 2.3.h 2.3.h 2.3.h
401 434 548 614 710 878 1070
6
2.3.h SLLL SLLL SLLL SLLL SLLL SLLL
673 725 908 1214
7
2.3.h 2.3.h SLLL SLLL
861 925 1054 1094 1617
8
2.3.h 2.3.h XXR SLLL 2.3.h
Table II. Known nontrivial lower bounds for higher two color
Ramsey numbers R (k , l ), with references.
Exoo in [Ex15] gives the bounds R (3, 27) ³ 158 and R (3, 31) ³ 198. The constructions
establishing R (3, 26) ³ 150, R (3, 29) ³ 174, R (3, 31) ³ 198 and R (3, 32) ³ 212 are presented in
[SLL1], [SLL3], [LSS] and [LSZL], respectively. Yu [Yu2] constructed a special class of
trianglefree cyclic graphs establishing several lower bounds for R (3, k ), for k ³ 61. Only two
of these bounds, R (3, 61) ³ 479 and R (3,103) ³ 955, cannot be easily improved by the inequality
R (3, 4k + 1) ³ 6R (3, k + 1)  5 from [CCD] (2.3.c) and data from Tables I and II. Finally,
for higher parameters we mention two more cases which improve on bounds listed in earlier
revisions: R (9, 17) ³ 1411 is given in [XXR] and R (10, 15) ³ 1265 can be obtained by using
2.3.h.
In general, one can expect that the lower bounds in Table II are weaker than those in
Table I, in the sense that with some work many of them should not be hard to improve, in
contrast to the bounds in Table I, especially smaller ones.
 6 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
2.3. Other results on R (k , l )
(a) R (k , l ) £ R (k 1, l ) + R (k , l 1), with strict inequality when both terms on the right hand
side are even [GG]. There are obvious generalizations of this inequality for avoiding
graphs other than complete.
(b) R (k , k ) £ 4R (k , k  2) + 2 [Walk].
(c) Explicit construction for R (3, 4k + 1) ³ 6R (3, k + 1)  5, for all k ³ 1 [CCD].
(d) Constructive results on trianglefree graphs in relation to the case of R (3, k ) [BBH1,
BBH2, Fra1, Fra2, FrLo, Gri, KM1, Loc, RK3, RK4, Stat, Yu1].
(e) Bounds for the difference between consecutive Ramsey numbers, in particular the bound
R (k , l ) ³ R (k , l  1) + 2k  3 for k , l ³ 3 [BEFS].
(f) By taking a disjoint union of two critical graphs one can easily see that R (k , p ) ³ s and
R (k , q ) ³ t imply R (k , p + q 1) ³ s + t 1. Xu and Xie [XX1] improved this construction
to yield better general lower bounds, in particular R (k , p + q 1) ³ s + t + k  3.
(g) For 2 £ p £ q and 3 £ k , if (k , p )graph G and (k , q )graph H have a common induced
subgraph on m vertices without Kk 1, then R (k , p + q  1)> n (G) + n (H) +m. In particular,
this implies the bounds R (k , p + q  1) ³ R (k , p ) + R (k , q ) + k  3 and
R (k , p + q  1) ³ R (k , p ) + R (k , q ) + p  2 [XX1, XXR].
(h) R (2k  1, l ) ³ 4R (k , l  1) 3 for l ³ 5 and k ³ 2, and in particular for k = 3 we obtain
R (5, l ) ³ 4R (3, l  1) 3 [XXER].
(i) If the quadratic residues Paley graph Qp of prime order p = 4t + 1 contains no Kk , then
R (k , k ) ³ p + 1 and R (k + 1, k + 1) ³ 2p + 3 [She1, Mat]. Data for larger p was
obtained in [LSL]. See also items 5.2.h and 5.2.i for similar multicolor results.
(j) Study of Ramsey numbers for large disjoint unions of graphs [Bu1, Bu9], in particular
R (nKk , nKl ) = n (k + l  1) + R (Kk 1,Kl 1)  2, for n large enough [Bu8].
(k) R (k , l ) ³ L (k , l ) + 1, where L (k , l ) is the maximal order of any cyclic (k , l )graph. A
compilation of many best cyclic bounds was presented in [HaKr].
(l) Twocolor lower bounds can be obtained by using items 5.2.k, 5.2.l and 5.2.m with
r = 2. Some generalizations of these were obtained in [ZLLS].
In the last six items of this section we only briefly mention some pointers to the literature
dealing with asymptotics of Ramsey numbers. This survey was designed mostly for small,
finite, and combinatorial results, but still we wish to give the reader some useful and representative
references to more traditional papers looking first of all at the infinite.
(m) In a 1995 breakthrough Kim proved that R (3, k ) = Q(k 2/ log k ) [Kim].
(n) Explicit trianglefree graphs with independence k on W(k 3/ 2 ) vertices [Alon2, CPR].
(o) Other general and asymptotic results on trianglefree graphs in relation to the case of
R (3, k ) [AKS, Alon2, CCD, CPR, Gri, FrLo, Loc, She2].
 7 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
(p) In 1947, Erdo¨s gave an amazingly simple probabilistic proof that R (k , k ) ³ c .k 2 k / 2
[Erd1]. Spencer [Spe1] improved the constant in the last result. More probabilistic
asymptotic lower bounds for other Ramsey numbers were obtained in [Spe1, Spe2,
AlPu].
(q) Other asymptotic bounds for R (k , k ) can be found, for example, in [Chu3, McS] (lower
bound) and [Tho] (upper bound), and for many other bounds in the general case of
R (k , l ) consult [Spe2, GRS, GrRo¨, Chu4, ChGra2, LRZ, AlPu, Kriv].
(r) Explicit construction of a graph with clique and independence k on 2c log2k / log log k vertices
by Frankl and Wilson [FraWi]. Further constructions by Chung [Chu3] and Grolmusz
[Grol1, Grol2]. Explicit constructions like these are usually weaker than known
probabilistic results.
3. Two Colors  Dropping One Edge from Complete Graph
H K3  e K4  e K5  e K6  e K7  e K8  e K9  e K10  e K11  e
G
K3  e 3 5 7 9 11 13 15 17 19
37 42
K3 5 7 11 17 21 25 31
38 47
29 34 41
K4  e 5 10 13 17 28
38
27 37
K4 7 11 19
36 52
31 40
K5  e 7 13 22
39 66
30 43
K5 9 16
34 67 112
31 45 59
K6  e 9 17
39 70 135
37
K6 11 21
55 119 205
40 59
K7  e 11 28
66 135 251
28 51
K7 13
34 88 204
Table III. Two types of Ramsey numbers R (G,H),
includes all known nontrivial values.
The exact values in Table III involving K3  e are trivial, since one can easily see that
R (K3  e ,Kk ) = R (K3  e ,Kk +1  e ) = 2k  1, for all k ³ 2. Other bounds (not shown in
Table III) can be obtained by using Table I, an obvious generalization of the inequality
 8 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
R (k , l ) £ R (k 1, l ) + R (k , l 1), and by monotonicity of Ramsey numbers, in this case
R (Kk 1,G) £ R (Kk  e ,G) £ R (Kk ,G). The upper bounds from the manuscripts [ShZ1,
ShZ2] are subsumed by a later article [Shi2].
H K4  e K5  e K6  e K7  e K8  e K9  e K10  e K11  e
G
MPR WWY2
K3 CH2 Clan FRS1 GH Ra1 Ra1
MPR MPR
Ea1 Ex14 Ex14
K4  e CH1 FRS2 McR McR
HZ2
Ex11 Ex14
K4 CH2 EHM1
Ea1 HZ2
Ex14 Ex14
K5  e FRS2 CEHMS
Ea1 HZ2
Ex8 Ea1
K5 BH
Ex8 HZ2 HZ2
Ex14 Ex14 Ex14
K6  e McR
Ea1 HZ2 HZ2
Ex14
K6 McN
Ea1 ShZ2 ShZ2
Ex14 Ex14
K7  e McR
HZ2 HZ2 ShZ1
Ea1 Ex14
K7 Ea1 ShZ2 ShZ2
References for Table III.
All (K3,Kl  e )graphs for l £ 6 have been enumerated [Ra1]. For the following
numbers it was established that the critical graphs are unique: R (K3,Kl  e ) for l = 3 [Tr], 6
and 7 [Ra1], R (K4  e ,K4  e ) [FRS2], R (K5  e ,K5  e ) [Ra3] and R (K4  e ,K7  e )
[McR]. The number of R (K3,Kl  e )critical graphs for l = 4, 5 and 8 is 4, 2 and 9, respectively
[MPR], and there are at least 6 such graphs for R (K3,K9  e ) [Ra1]. The bound
R (K3,K12  e ) ³ 46 is given in [MPR]. Wang, Wang and Yan in [WWY2] constructed
cyclic graphs showing R (K3,K13  e ) ³ 54, R (K3,K14  e ) ³ 59 and R (K3,K15  e ) ³ 69.
The upper bounds in [HZ2] were obtained by a reasoning generalizing the bounds for
classical numbers in [HZ1]. Several other results from section 2.3 apply, though checking in
which situation they do may require looking inside the proofs whether they still hold for
Kn  e .
 9 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
4. General Graph Numbers in Two Colors
This section includes data with respect to general graph results. We tried to include all
nontrivial values and identities regarding exact results (or references to them), but only those
out of general bounds and other results which, in our opinion, have a direct connection to the
evaluation of specific numbers. If some small value cannot be found below, it may be covered
by the cumulative data gathered in section 7, or be a special case of a general result listed in
this section. Note that B1 = F1 = C3 = W3 = K3, B2 = K4  e , P3 = K3  e , W4 = K4 and
C4 = K2,2 imply other identities not mentioned explicitly.
4.1. Paths
R (Pn , Pm ) = n + ëm/ 2 û  1 for all n ³m ³ 2 [GeGy]
4.2. Cycles
R (C3,C3) = 6 [GG]
R (C4,C4) = 6 [CH1]
Result obtained independently in [Ros] and [FS1], new simple proof in [Ka´Ros]:
R (Cn ,Cm ) =
ìïíïî
max{n  1 +m/ 2, 2m  1}
n  1 +m/ 2
2n  1
for 4 £m < n , m even and n odd
for 4 £m £ n , m and n even, (n ,m) =/ (4,4)
for 3 £m £ n , m odd, (n ,m) =/ (3,3)
Unions of cycles, formulas and bounds for R (nCp ,mCq ) [MS, Den]
R (nC3,mC3 ) = 3n + 2m for n ³m ³ 1, n ³ 2 [BES]
R (nC4,mC4 ) = 2n + 4m  1 for m ³ n ³ 1, (n ,m) =/ (1,1) [LiWa1]
Formulas for R (nC4,mC5 ) [LiWa2]
4.3. Wheels
R (W3,W5 ) = 11 [Clan]
R (W3,Wn ) = 2n 1 for all n ³ 6 [BE2]
All critical colorings for R (W3,Wn ) for all n ³ 3 [RaJi]
R (W4,W5 ) = 17 [He3]
R (W5,W5 ) = 15 [HaMe2, He2]
R (W4,W6 ) = 19, R (W5,W6 ) = 17 and R (W6,W6 ) = 17, and all critical colorings (2, 1
and 2) for these numbers [FM]. R (W6,W6 ) = 17 and c(W6 ) = 4 gives a counterexample
G =W6 to the Erdo¨s conjecture (see [GRS]) R (G,G) ³ R (Kc(G),Kc(G) ).
 10 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
4.4. Books
R (B1, Bn ) = 2n + 3 for all n>1 [RS1]
R (B3, B3) = 14 [RS1, HaMe2]
R (B2, B5) = 16, R (B3, B5) = 17, R (B5, B5) = 21,
R (B4, B4) = 18, R (B4, B6) = 22, R (B6, B6) = 26 [RS1]
254 £ R (B37, B88) £ 255 [Par6]
R (Bn , Bm ) = 2n + 3 for all n ³ cm for some c [NiRo1, NiRo2]
R (Bn , Bn ) = (4 + o (1))n [RS1, NiRS]
In general, R (Bn , Bn ) = 4n + 2 for 4n + 1 a prime power, and
several other general equalities and bounds for R (Bn , Bm ) [RS1, FRS7, Par6, NiRS].
4.5. Complete bipartite graphs
HINT: This section gathers information on Ramsey numbers where specific bipartite graphs
are avoided in a coloring of Kn (as everywhere in this survey), in contrast to often studied
bipartite Ramsey numbers (not covered in this survey) where the initial coloring is of a bipartite
graph Kn ,m .
R (K1,n ,K1,m ) = n +m  e, where e = 1 if both n and m are even and e = 0 otherwise
[Har1]. It is also a special case of multicolor numbers for stars obtained in [BuRo1].
R (nK1,3,mK1,3 ) = 4n +m  1 for n ³m ³ 1, n ³ 2 [BES]
R (K2,3,K2,3 ) = 10 [Bu4]
R (K2,3,K2,4 ) = 12 [ExRe]
R (K2,3,K1,7 ) = 13 [Par4]
R (K2,3,K3,3 ) = 13 and R (K3,3,K3,3) = 18 [HaMe3]
R (K2,2,K2,8 ) = 15 and R (K2,2,K2,11) = 18 [HaMe4]
R (K2,2,K1,15 ) = 20 [La2]
R (K2,n ,K2,n ) £ 4n  2 for all n ³ 2, exact values 6, 10, 14, 18, 21, 26, 30, 33, 38, 42,
46, 50, 54, 57 and 62 of R (K2,n ,K2,n ) for 2 £ n £ 16, respectively.
The first open diagonal case is 65 £ R (K2,17,K2,17 ) £ 66 [EHM2].
Conjecture that 4n  3 £ R (K2,n ,K2,n ) £ 4n  2 for n ³ 2 [LorMe1].
Bounds and some values for the numbers of the form R (Kk ,n ,Kk ,m ) [LorMe1], and
R (K2,n 1,K2,n ) and R (K2,n ,K2,n ) [LorMe2].
The values of R (K2,n ,K2,m ) for all 2 £ n ,m £ 10 are gathered in [LorMe3] except 8
cases, for which lower and upper bounds are given. Several theorems giving exact formulas
and bounds assuming special dependencies between n and m [LorMe3].
Asymptotics for K2,m versus Kn [CLRZ]
 11 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
Upper bound asymptotics for Kk ,m versus Kn [LZ]
See section 4.10 for stars versus various bipartite graphs
4.6. Triangle versus other graphs
R (3, k ) =Q(k 2/ log k ) [Kim]
Explicit construction for R (3, 4k + 1) ³ 6R (3, k + 1)  5, for all k ³ 1 [CCD]
Explicit trianglefree graphs with independence k on W(k 3/2 ) vertices [Alon2, CPR]
R (K3,K7  2P2 ) = R (K3,K7  3P2 ) = 18 [SchSch2]
R (K3,K3 + Km ) = R (K3,K3 + Cm ) = 2m + 5 for m ³ 212 [Zhou1]
R (K3,G) = 2n (G)  1 for any connected G on at least 4 vertices and with at most
(17n (G) + 1)/15 edges, in particular for G = Pi and G = Ci , for all i ³ 4 [BEFRS1]
R (K3,G) £ 2e (G) + 1 for any graph G without isolated vertices [Sid3, GK]
R (K3,G) £ n (G) + e (G) for all G, a conjecture [Sid2]
R (K3,G) for all connected G up to 9 vertices BBH1, BBH2], see also section 7.1
R (K3,Kn ), see section 2
R (K3,Kn  e ), see section 3
Formulas for R (nK3,mG) for all G of order 4 without isolates [Zeng]
Since B1 = F1 = C3 = W3 = K3, other sections apply
See also [AKS, BBH1, BBH2, FrLo, Fra1, Fra2, Gri, Loc, KM1, LZ, RK3, RK4, She2,
Spe2, Stat, Yu1]
4.7. Paths versus other graphs
P3 versus special graphs G [CH2]
Paths versus stars [Par2, BEFRS2]
Paths versus trees [FS4]
Paths versus books [RS2]
Paths versus cycles [FLPS, BEFRS2]
Paths versus Kn [Par1]
Paths versus Kn ,m [Ha¨g]
Paths versus W5 and W6 [SuBa1]
Paths versus W7 and W8 [Bas]
Paths versus wheels [BaSu, ChenZZ1]
Paths and cycles versus trees [FSS1]
Sparse graphs versus paths and cycles [BEFRS2]
Graphs with long tails [Bu2, BG]
Unions of paths [BuRo2]
 12 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
4.8. Cycles versus complete graphs
C3 C4 C5 C6 C7 C8 ... Cn for n ³m
6 7 9 11 13 15 2n  1
K3 GG CS CS FS1 FS1 FS1
...
FS1
9 10 13 16 19 22 3n  2
K4 GG CH2 He2/JR4 JR2 YHZ1 YHZ1
...
YHZ1
14 14 17 21 25 29 4n  3
K5 GG Clan He2/JR4 JR2 YHZ2 BJYHRZ
...
BJYHRZ
18 18 21 26 31 36 5n  4
K6 Ke´ry Ex2/RoJa1 JR5 Schi1 Schi1 Schi1
...
Schi1
23 22 25 37 43 6n  5
K7 Ka2/GY RT/JR1 Schi2 conj. conj.
...
conj.
28 26 50 7n  6
K8 GR/MZ RT conj.
...
conj.
36 ³ 30 8n  7
K9 Ka2/GR RT
...
conj.
40  43 ³ 34 9n  8
K10 Ex5/RK2 RT
...
conj.
Table IV. Known Ramsey numbers R (Cn ,Km ).
 The first column in Table IV gives data from the first row in Table I.
 Joint credit [He2/JR4] in Table IV refers to two cases in which Hendry [He2] announced
the values without presenting the proofs, which later were given in [JR4]. For other joint
credits in Table IV, the first reference is for the lower bound and the second for the upper
bound. The special cases of R (C6,K5 ) = 21 [JR2] and R (C7,K5 ) = 25 were also solved
independently in [YHZ2] and [BJYHRZ].
 Since 1976, it was conjectured that R (Cn ,Km ) = (n  1)(m  1) + 1 for all n ³ m ³ 3,
except n =m = 3 [FS4, EFRS2]. The parts of this conjecture were proved as follows: for
n ³ m2  2 [BoEr], for n > 3 = m [FS1], for n ³ 4 = m [YHZ1], for n ³ 5 = m
[BJYHRZ], for n ³ 6 = m [Schi1], for n ³ m ³ 7 with n ³ m(m  2) [Schi1], and for
n ³ 4m + 2, m ³ 3 [Nik]. Still open conjectured cases are marked in Table IV by "conj."
 General study of cycles versus Kn numbers, including asymptotics [BoEr, Spe2, FS4,
EFRS2, CLRZ, Sud1, ZaLi, AlRo¨].
 13 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
4.9. Cycles versus other graphs
C4 versus stars [Par3, Par5, BEFRS5, Chen, ChenJ, GoMC]
C4 versus trees [EFRS4, Bu7, BEFRS5, Chen]
C4 versus Km, n [HaMe4] and K2, n [LorMe3]
C4 versus all graphs on six vertices [JR3]
R (C4, Bn ) = 7, 9, 11, 12, 13 and 16, for 2 £ n £ 7, respectively [FRS6]
R (C4, Bn ) = 17, 18, 19, 20 and 21, for 8 £ n £ 12, respectively [Tse1]
R (C4, B13 ) = 22 and R (C4, B14 ) = 24 [Tse2]
R (C4,Wn ) = 10, 9, 10, 9, 11, 12, 13, 14, 16 and 17, for 4 £ n £ 13, respectively [Tse1]
R (C4,G) £ 2q + 1 for any isolatefree graph G with q edges [RoJa2]
R (C4,G) £ p + q  1 for any connected graph G on p vertices and q edges [RoJa2]
R (C5,W6 ) = 13 [ChvS]
R (C5,K6  e ) = 17 [JR4]
R (C5, B1 ) = R (C5, B2) = 9 [CRSPS]
R (C5, B3 ) = 10, and in general R (C5, Bn ) = 2n + 3 for n ³4 [FRS8]
C5 versus all graphs on six vertices [JR4]
R (C6,K5  e ) = 17 [JR2]
C6 versus all graphs on five vertices [JR2]
R (Cn ,G) £ 2q + ë n / 2 û  1, for 3 £ n £ 5, for any isolatefree graph G with q > 3 edges.
It is conjectured that it also holds for other n [RoJa2].
Cycles versus paths [FLPS, BEFRS2]
Cycles versus stars [La1, Clark, see Par6]
Cycles versus trees [FSS1]
Cycles versus books [FRS6, FRS8, Zhou1]
Cycles versus Kn ,m [BoEr]
Cycles versus W5 and W6 [SuBB2]
Cycles versus wheels [Zhou2]
See also bipartite graphs for K2,2 = C4
4.10. Stars versus other graphs
Stars versus C4 [Par3, Par5, Chen, ChenJ, GoMC]
Stars versus W5 and W6 [SuBa1]
Stars versus wheels [ChenZZ2]
Stars versus paths [Par2, BEFRS2]
Stars versus cycles [La1, Clark, see Par6]
Stars versus books [CRSPS, RS2]
Stars versus K2,n [Par4, GoMC]
Stars versus Kn ,m [Stev, Par3]
Stars versus bipartite graphs [Par4, Stev]
Stars versus trees [Bu1, Coc, GV, ZZ]
 14 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
Stars versus stripes [CL, Lor]
Stars versus Kn  tK 2 [Hua1, Hua2]
Stars versus 2K2 [MO]
Union of two stars [Gros2]
4.11. Books versus other graphs
R (B3,K4 ) = 14 [He3]
R (B3,K5 ) = 20 [He2][BaRT]
Books versus paths [RS2]
Books versus trees [EFRS7]
Books versus stars [CRSPS, RS2]
Books versus cycles [FRS6, FRS8, Zhou1, Tse1, Tse2]
Books versus Kn [LR1, Sud2]
Books versus wheels [Zhou3]
Books versus K2 + Cn [Zhou3]
Books and (K1 + tree ) versus Kn [LR1]
Generalized books Kr + qK1 versus Kn [NiRo3]
4.12. Wheels versus other graphs
R (W5,K5  e ) = 17 [He2][YH]
R (W5,K5 ) = 27 [He2][RST]
W5 and W6 versus stars and paths [SuBa1]
Wheels versus stars [ChenZZ2]
W5 and W6 versus trees [BSNM]
W5 and W6 versus cycles [SuBB2]
R (W6,C5 ) = 13 [ChvS]
W7 and W8 versus paths [Bas]
W7 versus trees T with D(n (T )) ³ n (T )  3 [ChenZZ3]
Wheels versus paths [BaSu, ChenZZ1]
Odd wheels versus starlike trees [SuBB1]
Wheels versus C4 [Tse1]
Wheels versus cycles [Zhou2]
Wheels versus books [Zhou3]
Wheels versus linear forests [SuBa2]
4.13. Trees and Forests
Trees, forests [Bu1, Bu7, CsKo, EFRS3, EG, FSS1, GeGy, GHK, GRS, GV, HaŁT]
Trees versus Kn [Chv]
Trees versus C4 [EFRS4, Bu7, Chen]
Trees versus paths [FS4]
Trees versus paths and cycles [FSS1]
Trees versus stars [Bu1, Coc, GV, ZZ]
 15 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
Trees versus books [EFRS7]
Trees versus W5 and W6 [BSNM]
Trees T with D(n (T )) ³ n (T )  3 versus W7 [ChenZZ3]
Starlike trees versus odd wheels [SuBB1, ChenZZ3]
Trees versus Kn + Km [RS2, FSR]
Trees versus bipartite graphs [BEFRS5, EFRS6]
Trees versus almost complete graphs [GJ2]
Trees versus small (n (G) £ 5) connected G [FRS4]
Trees versus multipartite complete graphs [EFRS8, BEFRSGJ]
Linear forests, forests [BuRo2, FS3, CsKo]
Linear forests versus wheels [SuBa2]
Forests versus Kn [Stahl]
Forests versus almost complete graphs [CGP]
4.14. Mixed special cases:
R (C5 + e ,K5 ) = 17 [He5]
R (W5,K5  e ) = 17 [He2][YH]
R (B3,K5 ) = 20 [He2][BaRT]
R (W5,K5 ) = 27 [He2][RST]
25 £ R (K5  P3,K5 ) £ 28 [He2]
26 £ R (K2,2,2 ,K2,2,2 ), K2,2,2 is an octahedron [Ex8]
4.15. Mixed general cases
Unicyclic graphs [Gros1, Ko¨h, KrRod]
K2,m and C2m versus Kn [CLRZ]
K2,n versus any graph [RoJa2]
nK3 versus mK3, in particular R (nK3, nK3 ) = 5n for n ³ 2 [BES]
nK3 versus mK4 [LorMu]
R (nK4, nK4 ) = 7n + 4 for large n [Bu8]
2K2 versus Kn and general graphs G [CH2]
Variety of results on numbers R (nG ,mH) [Bu1]
Stripes [CL, Lor]
Union of two stars [Gros2]
Double stars* [GHK]
Graphs with bridge versus Kn [Li]
Fans Fn = K1 + nK2 versus Km [LR2]
R (F1, Fn ) = R (K3, Fn ) = 4n + 1 for n ³ 2 , and bounds for R (Fm , Fn ) [GGS]
Multipartite complete graphs [BEFRS3, EFRS4, FRS3, Stev]
* double star is a union of two stars with their centers joined by an edge
 16 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
Multipartite complete graphs versus trees [EFRS8, BEFRSGJ]
Disconnected graphs versus any graph [GJ1]
Graphs with long tails [Bu2, BG]
Brooms+ [EFRS3]
4.16. Other general results
[Chv] R (Kn , Tm ) = (n 1)(m1) + 1 for any tree T on m vertices.
[CH2] R (G,H) ³ ( c(G)  1)(c (H)  1) + 1, where c(G) is the chromatic number of
G, and c (H) is the size of the largest connected component of H.
[BE1] R (G,G) ³ ë(4n (G)  1) / 3û for any connected G, and R (G,G) ³ 2n  1 for
any connected nonbipartite G.
[BE2] Graphs yielding R (Kn ,G) = (n 1)(n (G)  1) + 1 and related results (see also
[EFRS5]).
[Bu2] Graphs H yielding R (G,H) = (c(G)  1)(n (H)  1) + s (G), where s (G) is a
chromatic surplus of G, defined as the minimum number of vertices in some
color class under all vertex colorings in c(G) colors (such H’s are called Ggood).
This idea, initiated in [Bu2], is a basis of a number of exact results for
R (G,H) for large and sparse graphs H [BG, BEFRS2, BEFRS4, Bu5, FS,
EFRS4, FRS3, BEFSRGJ, BF, LR4]. A survey of this area appeared in
[FRS5].
[BaLS] Graph G is Ramsey saturated if R (G + e ,G + e ) > R (G,G) for every edge e
in G. Several theorems on Ramsey saturated and unsaturated graphs. A conjecture
that almost all graphs are Ramsey unsaturated.
[Par3] Relations between some Ramsey graphs and block designs. See also [Par4].
[Bra3] R (G,H) > h (G, d ) n (H) for all nonbipartite G and almost every d regular
H, for some h unbounded in d .
[LZ] Lower bound asymptotics of R (G,H) for large dense H [LZ].
[CSRT] R (G,G) £ cd n (G) for all G, where constant cd depends only on the maximum
degree d in G. The constant was improved in [GRR1]. Tight lower and
upper bounds for bipartite G [GRR2].
[ChenS] R (G,G) £ cd n for all d arrangeable graphs G on n vertices, in particular
with the same constant for all planar graphs. The constant cd was improved in
[Eaton]. An extension to graphs not containing a subdivision of Kd [Ro¨Th].
Progress towards a conjecture that the same inequality holds for all d 
degenerate graphs G [KoRo¨1, KoRo¨2, KoSu].
[EFRS9] Study of graphs G, called Ramsey size linear, for which there exists a constant
cG such that for all H with no isolates R (G,H) £ cG e (H). An overview and
+ broom is a star with a path attached to its center
 17 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
further results were given in [BaSS].
[LRS] R (G,G) < 6n for all n vertex graphs G, in which no two vertices of degree
at least 3 are adjacent. This improves the result R (G,G) £ 12n in [Alon1].
[AlKS] Discussion of a conjecture by Erdo¨s that there exists a constant c such that
R (G,G) £ 2c Öe (G). Proof for bipartite graphs G and progress towards the
conjecture in other cases.
[Kriv] Lower bound on R (G,Kn ) depending on the density of subgraphs of G. This
construction for G = Km produces a bound similar to the best known probabilistic
lower bound by Spencer [Spe2].
[NiRo3] R (Kp + 1, Bq
r ) = p (q + r  1) + 1 for generalized books Bq
r = Kr + qK1, for all
sufficiently large q .
[Shi1] R (Qn ,Qn ) £ 2(3 + Ö5)n / 2 + o (n ), for the n dimensional cube Qn with 2n vertices.
This bound can also be derived from a theorem in [KoRo¨1].
[Gros1] Conjecture that R (G,G) = 2n (G)  1 if G is unicyclic of odd girth. Further
support for the conjecture was given in [Ko¨h, KrRod].
[RoJa2] R (K2,k ,G) £ kq + 1, for k ³ 2, for isolatefree graphs G with q ³ 2 edges.
[FSS1] Discussion of the conjecture that R (T1, T2) £ n (T1) + n (T2)  2 holds for all
trees T1, T2. See also [Bu1, Bu7, CsKo, EFRS3, EG, GeGy, GHK, GRS, GV].
[HaŁT] If tree T is viewed as a bipartite graph with parts t 1 and t 2, t 2 ³ t 1, let
b (T ) = max(2t 1 + t 2  1, 2t 2  1). Then the bound R (T , T ) ³ b (T ) holds
always, and R (T , T ) = b (T ) holds for many classes of trees, and asymptotically.
[FM] R (W6,W6 ) = 17 and c(W6 ) = 4. This gives a counterexample G =W6 to the
Erdo¨s conjecture (see [GRS]) R (G,G) ³ R (Kc(G),Kc(G) ).
[LR3] Bounds on R (H + Kn ,Kn ) for general H. Also, for fixed k and m, as n® ¥,
R (Kk + Km ,Kn ) £ (m + o (1)) n k / (log n )k 1 [LRZ].
[Zeng] Formulas for R (nK3,mG) for all isolatefree graphs G on 4 vertices.
[BES] Study of Ramsey numbers for multiple copies of graphs.
See also [Bu1, Bu8, Bu9, LorMu].
[HaKr] Study of cyclic graphs yielding lower bounds for Ramsey numbers. Exact formulas
for paths and cycles, small complete graphs and for graphs with up to
five vertices.
[Bu6] Given integer m and graphs G and H, determining whether R (G,H) £ m
holds is NPhard.
[] Special cases of multicolor results listed in section 5.
[] See also surveys listed in section 7.
 18 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
5. Multicolor Graph Numbers
The only known value of a multicolor classical Ramsey number:
R3(3) = R (3,3,3) = R (3,3,3 ; 2) = 17 [GG]
2 critical colorings (on 16 vertices) [KaSt, LayMa]
2 colorings on 15 vertices [Hein]
115 colorings on 14 vertices [PR1]
General upper bound, implicit in [GG]:
R (k 1, ... , kr ) £ 2  r +
i = 1
Sr
R (k 1, ... , ki  1, ki  1, ki + 1, ... , kr ) (a)
Inequality in (a) is strict if the right hand side is even, and at least one of the terms in the
summation is even. It is suspected that this upper bound is never tight for r ³ 3 and ki ³ 3,
except for r = k 1 = k 2 = k 3 = 3. However, only two cases are known to improve over (a),
namely R4(3) £ 62 [FKR] and R (3,3,4) £ 31 [PR1, PR2], for which (a) produces only the
bounds of 66 and 34, respectively.
5.1. Bounds for multicolor classical numbers
Diagonal Cases
m 3 4 5 6 7 8 9
r
17 128 415 1070 3214 5384 13761
3
GG HiIr XXER Mat Xu XX2 XXER
51 634 3049 15202 62017
4
Chu1 XXER Xu XXER XXER
162 3416 26912
5
Ex10 XXER Xu
538
6
FreSw
1682
7
FreSw
Table V. Known nontrivial lower bounds for diagonal multicolor
Ramsey numbers Rr (m), with references.
The best published bounds corresponding to the entries in Table V marked by personal communication
[Xu] are: 3211 £ R3(7) [Mat], 2721 £ R4(5) [XXER] and 26082 £ R5(5) [XXER].
 19 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
The most studied and intriguing open case is
[Chu1] 51 £ R4(3) = R (3,3,3,3) £ 62 [FKR]
The inequality 5.a implies R4(3) £ 66, Folkman [Fo] in 1974 improved this bound to 65,
and Sa´nchezFlores [San] in 1995 proved R4(3) £ 64. The upper bounds in
162 £ R5(3) £ 307, 538 £ R6(3) £ 1838, 1682 £ R7(3) £ 12861, and 128 £ R (4, 4, 4) £ 236
are implied by 5.(a) (we repeat lower bounds from Table V just to see easily the ranges).
OffDiagonal Cases
Three colors:
m 4 5 6 7 8 9 10 11 12 13 14
k
30 45 60 79 98 110 141 157 181 205 233
3
Ka2 Ex2 Rob3 Ex16 ZSL SLZL 5.2.c 5.2.c 5.2.c 5.2.c 5.2.c
55 80 99
4
KLR Ex12 5.2.g
80 123
5
Ex12 5.2.g
Table VI. Known nontrivial lower bounds for 3color
Ramsey numbers of the form R (3, k ,m), with references.
In addition, the bounds 303 £ R (3,6,6), 609 £ R (3,7,7) and 1689 £ R (3,9,9) were derived in
[XXER] (used there for building other lower bounds for some diagonal cases).
The other most studied, and perhaps the only open case of a classical multicolor Ramsey
number, for which we can anticipate exact evaluation in the nottoodistance future is
[Ka2] 30 £ R (3,3,4) £ 31 [PR1, PR2]
In [PR1] it is conjectured that R (3,3,4) = 30, and the results in [PR2] eliminate some
cases which could give R (3,3,4) = 31. The upper bounds in 45 £ R (3,3,5) £ 57,
55 £ R (3,4,4) £ 79, and 80 £ R (3,4,5) £ 160 are implied by 5.(a) (we repeat lower bounds
from the Table VI to show explicitly the current ranges).
Four colors:
93 £ R (3,3,3,4) £ 153 [Ex16, XXER], 5.(a)
162 £ R (3,3,3,5) [XXER]
171 £ R (3,3,4,4) [Ex16, XXER]
561 £ R (3,3,3,11) [XX2, XXER]
 20 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
Lower bounds for higher numbers can be obtained by using general constructive results
from section 5.2 below. For example, the bounds 193 £ R (3,4,8), 261 £ R (3,3,15) and
241 £ R (3,3,3,7) were not published explicitly but are implied by 5.2.(c), 5.2.(c) and 5.2.(d),
respectively.
5.2. General multicolor results for complete graphs
(b) Rr (3) ³ 3Rr 1(3) + Rr 3(3)  3 [Chu1]
(c) R (3, k , l ) ³ 4R (k , l  1) 3 , and in general for r ³ 2 and ki ³ 2
R (3, k 1, ... , kr ) ³ 4R (k 1  1, k 2, ... , kr )  3 for k 1 ³ 5, and
R (k 1, 2k 2  1, k 3, ... , kr ) ³ 4R (k 1  1, k 2, ... , kr )  3 for k 1 ³ 5 [XX2, XXER]
(d) R (3, 3, 3, k 1, ... , kr ) ³ 3R (3, 3, k 1, ... , kr ) + R (k 1, ... , kr )  3 [Rob2]
(e) Bounds for Rk (3) [AbbH, Fre, Chu2, ChGri, GrRo¨, Wan]
(f) R (k 1, ... , kr ) ³ S (k 1, ... , kr ) + 2, where S (k 1, ... , kr ) is the generalized Schur number
[AbbH, Gi1, Gi2]. In particular, the special case k 1 = ... = kr = 3 has been widely studied
[Fre, FreSw, Ex10, Rob3].
(g) R (k 1, ... , kr ) ³ L (k 1, ... , kr ) + 1, where L (k 1, ... , kr ) is the maximal order of any cyclic
(k 1, ... , kr )coloring, which can be considered a special case of Schur partitions defining
(symmetric) Schur numbers. Many lower bounds for Ramsey numbers were established
by cyclic colorings. The following recurrence can be used to derive lower bounds for
higher parameters. For ki ³ 3
L (k 1, ... , kr , kr + 1 ) ³ (2kr + 1  3)L (k 1, ... , kr )  kr + 1 + 2 [Gi2]
(h) Rr (m) ³ p + 1 and Rr (m + 1) ³ r ( p + 1) + 1 if there exists a Km free cyclotomic r  class
association scheme of order p [Mat].
(i) If the quadratic residues Paley graph Qp of prime order p = 4t + 1 contains no Kk , then
R (s , k + 1, k + 1) ³ 4ps 6p + 3 [XXER].
(j) Rr (m) ³ cm (2m  3)r , and some slight improvements of this bound for small values of m
[AbbH, Gi1, Gi2, Song2].
(k) Rr ( pq + 1)> (Rr ( p + 1)  1)(Rr (q + 1)  1) [Abb1]
(l) Rr ( pq + 1)> Rr ( p + 1)(Rr (q + 1)  1) for p ³ q [XXER]
(m) R ( p 1q 1+ 1, ... , pr qr + 1) > (R ( p 1+ 1, ... , pr + 1)  1)(R (q 1+ 1, ... , qr + 1)  1) [Song3]
(n) Rr + s (m)> (Rr (m)  1)(Rs (m)  1) [Song2]
(o) R (k 1, k 2, ... , kr ) > (R (k 1, ... , ki )  1)(R (ki +1, ... , kr )  1) in [Song1], see [XXER].
(p) R (k 1, k 2, ... , kr ) > (k 1 + 1)(R (k 2  k 1 + 1, k 3, ... , kr )  1) [Rob4]
 21 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
(q) Further lower bound constructions, though with more complicated assumptions, were
presented in [XX2, XXER].
(r) Grolmusz [Grol1] generalized the classical constructive lower bound by Frankl and Wilson
[FraWi] (section 2.3.r) to more colors and to hypergraphs [Grol3] (section 6).
All lower bounds in (b) through (r) above are constructive. (d) generalizes (b), (m) generalizes
both (k) and (o), and (o) generalizes (n). (l) is stronger than (k). Finally observe that
the construction (m) with q 1 = ... = qi = 1 = pi +1 = ... = pr is the same as (o).
5.3. Special multicolor cases
R3(C4 ) = 11 [BS, see also Clap]
R3(C5 ) = 17 [YR1]
R3(C6 ) = 12 [YR2]
R3(C7 ) = 25 [FSS2]
18 £ R4(C4 ) £ 19 [Ex2] [Eng]
27 £ R5(C4 ) £ 29 [LaWo1]
R (C4,C4,K3 ) = 12 [Schu]
R (C4,K3,K3 ) = 17 [ExRe]
13 £ R (C3,C4,C5 ) [Rao]
R (K1,3,C4,K4 ) = 16 [KM2]
R (P4, P4,C3 ) = 9 [AKM]
R (P4, P4,C4 ) = 7 [AKM]
R (P4, P4,C5 ) = 9 [DzKu]
R (K4  e ,K4  e , P3 ) = 11 [Ex7]
28 £ R3(K4  e ) £ 30 [Ex7] [Piw2]
R (C4,C4,C4, T ) = 16 for T = P4 and T = K1,3 [ExRe]
27 £ R (K3,K3,C4,C4 ) [Eng]
86 £ R (K4,K4,C4,C4 ) [Bev], 5.2.(o)+
All colorings for (K4  e ,K4  e , P3 ) were found in [Piw2].
5.4. General multicolor results for cycles and paths
 R (Cn ,Cn ,Cn ) £ (4 + o (1)) n , with equality for odd n [Łuc]. It was conjectured by
Bondy and Erdo¨s, see [Erd2], that R (Cn ,Cn ,Cn ) £ 4n  3 for n ³ 4. If true, then
for all odd n ³ 5 we have R (Cn ,Cn ,Cn ) = 4n  3.
 Formulas for R (Cn ,Cm ,Ck ) and R (Cn ,Cm ,Ck ,Cl ) for n sufficiently large [EFRS1].
 22 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
 Rk (C4 ) £ k 2 + k + 1 for all k ³ 1, Rk (C4 ) ³ k 2  k + 2 for all k  1 which is a
prime power [Ir, Chu2, ChGra1], and Rk (C4 ) ³ k 2 + 2 for odd prime power k
[LaWo1]. The latter was extended to any prime power k in [Ling, LaMu].
 Bounds for Rk (Cn ) [Bu1, GRS].
 R (P3,Cn ,Cn ) = 2n  1 ( = R (Cn ,Cn ) ) for odd n ³ 5 [DzKu].
 R (P4, P4,Cn ) = n + 2 for n ³ 6, and R (P3, P5,Cn ) = n + 1 for n ³ 8 [DzKu].
 Formulas for Rk (P3 ) for all k , and for Rk (P4 ) if k is not divisible by 3 [Ir]. Wallis
[Wall] showed R6(P4 ) = 13, which already implied R3t (P4 ) = 6t + 1, for all t ³ 2.
Independently, the case Rk (P4 ) for k =/ 3m was completed by Lindstro¨m in [Lind],
and later Bierbrauer proved R3m (P4 ) = 2.3m + 1 for all m ³ 1.
 Monotone paths and cycles [Lef].
 Formulas for R (Pn1
, ... , Pnk
), except few cases [FS2].
 Formulas for R (n 1P2, ... , nk P2 ) [CL1].
 Formulas for R (pP3, qP3, rP 3 ) and R (pP4, qP4, rP 4 ) [Scob].
 See also sections 5.3 and 7.2, especially [AKM] for a number of small cases in three
colors similar to those listed in section 5.3.
 Study of asymptotics for R (Cm , ... ,Cm ,Kn ) [AlRo¨].
 Study of asymptotics for R (C2m ,C2m ,Kn ) for fixed m [ShiuLL, AlRo¨].
5.5. Other general multicolor results
 General bounds for Rk (G) [CH3, Par6].
 Formulas for Rk (G) for G being one of P3, 2K2 and K1,3 for all k , and for P4 if k
is not divisible by 3 [Ir].
 Bounds on Rk (Ks , t ), in particular for K2,2 = C4 and K2, t [ChGra1, AFM].
 tk 2 + 1 £ Rk (K2, t +1) £ tk 2 + k + 2, where the upper bound is general, and the lower
bound holds when both t and k are prime powers [ChGra1, LaMu].
 Bounds on Rk (G) for unicyclic graphs G of odd girth. Some exact values for special
graphs G, for k = 3 and k = 4 [KrRod].
 Formulas for R (S 1, ... , Sk ), where Si ’s are arbitrary stars [BuRo1].
 Formulas for R (S 1, ... , Sk ,Kn ), where Si ’s are arbitrary stars [Jac].
 Formulas for R (S 1, ... , Sk , nP2), where Si ’s are arbitrary stars [CL2].
 Formulas for R (S 1, ... , Sk , T ), where Si ’s are stars and T is a tree [ZZ].
 Study of R (G1, ... ,Gk ,G) for large sparse G [EFRS1, Bu3].
 Study of asymptotics for R (Cn , ... ,Cn ,Km ) [AlRo¨].
 Cockayne and Lorimer [CL1] found the exact formula for R (n 1P2, ... , nk P2), and
later Lorimer [Lor] extended it to a more general case of R (Km , n 1P2, ... , nk P2).
 23 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
Still more general cases of the latter, with multiple copies of the complete graph and
forests, were studied in [Stahl, LorSe, LorSo].
 If G is connected and R (Kk ,G) = (k 1)(n (G)  1) + 1, in particular if G is any tree,
then R (Kk 1
, ... ,Kkr
,G) = (R (k 1, ... , kr )  1)(n (G)  1) + 1 [BE2]. A generalization for
connected G1, ... ,Gn in place of G appeared in [Jac].
 If F ,G,H are connected graphs then R (F ,G,H) ³ (R (F ,G)  1)(c(H)  1) +
min{ R (F ,G), s (H) }, where s (G) is the chromatic surplus of G (see item [Bu2] in
section 4.16). This leads to several formulas and bounds for F and G being stars
and/or trees when H = Kn [ShiuLL].
 R (Kk 1
, ... ,Kkr
,G1, ... ,Gs ) ³ (R (k 1, ... , kr )  1)(R (G1, ... ,Gs )  1) for arbitrary graphs
G1, ... ,Gs [Bev]. This generalizes 5.2.(o).
 Constructive bound R (G1, ...,Gt n 1 ) ³ t n + 1 for some families of decompositions of
Kt n [LaWo1, LaWo2].
 Bounds for trees Rk (T ) and forests Rk (F ) [EG, GRS, BB, GT, Bra1, Bra2, SwPr].
 Bounds on Rk (G) for trees, forests, stars and cycles [Bu1].
 See also surveys listed in section 7.
6. Hypergraph Numbers
The only known value of a classical Ramsey number for hypergraphs:
R (4,4 ; 3) = 13 [MR1]
more than 200000 critical colorings
Other hypergraph cases:
33 £ R (4, 5 ; 3) [Ex13]
63 £ R (5, 5 ; 3) [Ea1]
56 £ R (4,4,4 ; 3) [Ex8]
34 £ R (5, 5 ; 4) [Ex11]
R (K4  t ,K4  t ; 3) = 7 [Ea2]
R (K4  t ,K4 ; 3) = 8 [Sob, Ex1, MR1]
14 £ R (K4  t ,K5 ; 3) [Ex1]
13 £ R (K4  t ,K4  t ,K4  t ; 3) £ 17 [Ex1] [Ea1]
The computer evaluation of R (4,4 ; 3) in [MR1] consisted of an improvement of the
upper bound from 15 to 13, which followed an extensive theoretical study of this number in
 24 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[Gi4, Is1, Sid1]. Exoo in [Ex1] announced the bounds R (4, 5 ; 3) ³ 30 and R (5, 5 ; 4) ³ 27
without presenting the constructions. The bound of R (4, 5 ; 3) ³ 24 was obtained by Isbell
[Is2]. Shastri in [Sha] shows a weak bound R (5, 5 ; 4) ³ 19 (now 34 in [Ex11]), nevertheless
his lemmas and those in [Ka3, Abb2, GRS, HuSo] can be used to derive other lower bounds
for higher numbers.
General hypergraph results:
 Several lower bound constructions for 3uniform hypergraphs were presented in [HuSo].
Study of lower bounds on R ( p , q ; 4) can be found in [Song3] and [SYL, Song4] (the
latter two papers are almost the same in contents). Most lower bounds in these papers can
be easily improved by using the same techniques, but starting with better constructions for
small parameters listed above.
 Let H (r )(s , t ) be the complete r partite r uniform hypergraph with r  2 parts of size 1,
one part of size s , and one part of size t (for example, for r = 2 it is the same as Ks , t ).
For the multicolor numbers, Lazebnik and Mubayi [LaMu] proved that
tk 2  k + 1 £ Rk (H (r )(2, t +1)) £ tk 2 + k + r ,
where the lower bound holds when both t and k are prime powers. For the general case
of H (r )(s , t ), more bounds are presented in [LaMu].
 Grolmusz [Grol1] generalized the classical constructive lower bound by Frankl and Wilson
[FraWi] (section 2.3.r) to more colors and to hypergraphs [Grol3].
 Lower bounds on Rm (k ; s ) are discussed in [DLR, AbbW]. In [AbbS], it is shown that
for some values of a , b the numbers R (m, a , b ; 3) are at least exponential in m.
 General lower bounds for large number of colors were given in an early paper by Hirschfeld
[Hir], and some of them were later improved in [AbbL].
 Other theoretical results on hypergraph numbers are gathered in [GrRo¨, GRS].
7. Cumulative Data and Surveys
7.1. Cumulative data for two colors
[CH1] R (G,G) for all graphs G without isolates on at most 4 vertices.
[CH2] R (G,H) for all graphs G and H without isolates on at most 4 vertices.
[Clan] R (G,H) for all graphs G on at most 4 vertices and H on 5 vertices, except
five entries (now all solved).
[He4] All critical colorings for R (G,H), for isolatefree graphs G and H as in
[Clan] above.
[Bu4] R (G,G) for all graphs G without isolates and with at most 6 edges.
[He1] R (G,G) for all graphs G without isolates and with at most 7 edges.
 25 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[HaMe2] R (G,G) for all graphs G on 5 vertices and with 7 or 8 edges.
[He2] R (G,H) for all graphs G and H on 5 vertices without isolates, except 7
entries (3 still open, see the paragraph at the end of this section).
[HoMe] R (G,H) for G = K1,3 + e and G = K4  e versus all connected graphs H on 6
vertices, except R (K4  e ,K6 ). The result R (K4  e ,K6 ) = 21 was claimed by
McNamara [McN, unpublished].
[FRS4] R (G, T ) for all connected graphs G on at most 5 vertices and all (except some
cases) trees T .
[FRS1] R (K3,G) for all connected graphs G on 6 vertices.
[Jin] R (K3,G) for all connected graphs G on 7 vertices. Some errors in [Jin] were
found by [SchSch1].
[Brin] R (K3,G) for all connected graphs G on at most 8 vertices. The numbers for
K3 versus sets of graphs with fixed number of edges, on at most 8 vertices,
were presented in [KM1].
[BBH1] R (K3,G) for all connected graphs G on 9 vertices. See also [BBH2].
[JR3] R (C4,G) for all graphs G on at most 6 vertices.
[JR4] R (C5,G) for all graphs G on at most 6 vertices.
[JR2] R (C6,G) for all graphs G on at most 5 vertices.
[LorMe3] R (K2,n ,K2,m ) for all 2 £ n ,m £ 10 except 8 cases, for which lower and upper
bounds are given.
[HaKr] All best lower bounds up to 102 from cyclic graphs. Formulas for best cyclic
lower bounds for paths and cycles, small complete graphs and for graphs with
up to five vertices.
Chva´tal and Harary [CH1, CH2] formulated several simple but very useful observations
how to discover values of some numbers. All five missing entries in the tables of Clancy
[Clan] have been solved. Out of 7 open cases in [He2] 4 have been solved, namely
R (4, 5) = R (G19,G23 ) = 25 and the items 2, 3 and 4 in section 4.14. The still open 3 cases
are for K5 versus the graphs K5 (section 2.1), K5  e (section 3), and K5  P3 (section 4.14).
7.2. Cumulative data for three colors
[YR3] R3(G) for all graphs G with at most 4 edges and no isolates.
[YR1] R3(G) for all graphs G with 5 edges and no isolates, except K4  e .
The case of R3(K4  e ) remains open (see section 5.3).
[YY] R3(G) for all graphs G with 6 edges and no isolates, except 10 cases.
[AKM] R (F ,G,H) for most triples of isolatefree graphs with at most 4 vertices.
Some of the missing cases completed in [KM2].
 26 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
7.3. Surveys
[Bu1] A general survey of results in Ramsey graph theory by S. A. Burr (1974)
[Par6] A general survey of results in Ramsey graph theory by T. D. Parsons (1978)
[Har2] Summary of progress by Frank Harary (1981)
[ChGri] A general survey of bounds and values by F. R. K. Chung and C. M. Grinstead
(1983)
[JGT] Special volume of the Journal of Graph Theory (1983)
[Rob1] A review of Ramsey graph theory for newcomers by F. S. Roberts (1984)
[Bu7] What can we hope to accomplish in generalized Ramsey Theory ? (1987)
[GrRo¨] Survey of asymptotic problems by R. L. Graham and V. Ro¨dl (1987)
[GRS] An excellent book by R. L. Graham, B. L. Rothschild and J. H. Spencer,
second edition (1990)
[FRS5] Survey by Faudree, Rousseau and Schelp of graph goodness results, i.e. conditions
for the formula R (G,H) = ( c(G)  1 ) ( n (H)  1 ) + s (G) (1991)
[Nes˘] A chapter in Handbook of Combinatorics by J. Nes˘etr˘il (1996)
[Caro] Survey of zerosum Ramsey theory by Y. Caro (1996)
[Chu4] Among 114 open problems and conjectures of Paul Erdo¨s, presented and commented
by F. R. K. Chung, 31 are concerned directly with Ramsey numbers.
216 references are given (1997). An extended version of this work was
prepared jointly with R. L. Graham [ChGra2]. (1998)
[CoPC] Special issue of Combinatorics, Probability and Computing (2003)
The surveys by S. A. Burr [Bu1] and T. D. Parsons [Par6] contain extensive chapters on
general exact results in graph Ramsey theory. F. Harary presented the state of the theory in
1981 in [Har2], where he also gathered many references including seven to other early surveys
of this area. More than two decades ago, Chung and Grinstead in their survey paper
[ChGri] gave less data than in this work, but included a broad discussion of different
methods used in Ramsey computations in the classical case. S. A. Burr, one of the most
experienced researchers in Ramsey graph theory, formulated in [Bu7] seven conjectures on
Ramsey numbers for sufficiently large and sparse graphs, and reviewed the evidence for them
found in the literature. Three of them have been refuted in [Bra3].
For newer extensive presentations see [GRS, GrRo¨, FRS5, Nes˘, Chu4, ChGra2], though
these focus on asymptotic theory not on the numbers themselves. Finally, this compilation
could not pretend to be complete without mentioning special volumes of the Journal of Graph
Theory [JGT, 1983] and Combinatorics, Probability and Computing [CoPC, 2003], dedicated
entirely to Ramsey theory. Besides a number of research papers, they include historical notes
and present to us Frank P. Ramsey (19031930) as a person.
 27 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
8. Concluding Remarks
This compilation does not include information on numerous variations of Ramsey
numbers, nor related topics, like size Ramsey numbers, zerosum Ramsey numbers, irredundant
Ramsey numbers, induced Ramsey numbers, local Ramsey numbers, connected Ramsey
numbers, chromatic Ramsey numbers, avoiding sets of graphs in some colors, coloring graphs
other than complete, or the so called Ramsey multiplicities. Interested reader can find such
information in the surveys listed in section 7 here.
The author apologizes for any omissions or other errors in reporting results belonging to
the scope of this work. Suggestions for any kind of corrections or additions will be greatly
appreciated and considered for inclusion in the next revision of this survey.
Acknowledgement
I would like to thank Brendan McKay, Geoffrey Exoo and Heiko Harborth for their help
in gathering data for earlier versions of this survey.
References
We mark the papers containing results obtained with the help of computer algorithms
with stars. We identify two categories of such papers: marked with * involving some use of
computers, where the results are easily verifiable with some computations, and those marked
with **, where cpu intensive algorithms have to be implemented to replicate or verify the
results. The first category contains mostly constructions done by algorithms, while the second
mostly nonexistence results or claims of complete enumerations of special classes of graphs.
The references are ordered alphabetically by the last name of the first author, for the
same first author by the last name of the second author, etc. We preferred that all work by the
same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations
are not in alphabetical order, for example [BaRT] is earlier on the list than [BaLS].
[Abb1] H.L. Abbott, Ph. D. thesis, University of Alberta, Edmonton, 1965.
[Abb2] H.L. Abbott, A Theorem Concerning Higher Ramsey Numbers, in Infinite and Finite Sets, (A. Hajnal,
R. Rado and V.T. So´s eds.) North Holland, (1975) 2528.
[AbbH] H.L. Abbott and D. Hanson, A Problem of Schur and Its Generalizations, Acta Arithmetica, 20
(1972) 175187.
[AbbL] H.L. Abbott and Andy Liu, Remarks on a Paper of Hirschfeld Concerning Ramsey Numbers,
Discrete Mathematics, 39 (1982) 327328.
[AbbS] H.L. Abbott and M.J. SmugaOtto, Lower Bounds for Hypergraph Ramsey Numbers, Discrete
Applied Mathematics, 61 (1995) 177180.
[AbbW] H.L. Abbott and E.R. Williams, Lower Bounds for Some Ramsey Numbers, Journal of Combinatorial
Theory, Series A, 16 (1974) 1217.
[AKS] M. Ajtai, J. Komlo´s and E. Szemere´di, A Note on Ramsey Numbers, Journal of Combinatorial
Theory, Series A, 29 (1980) 354360.
 28 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[Alon1] N. Alon, Subdivided Graphs Have Linear Ramsey Numbers, Journal of Graph Theory, 18 (1994)
343347.
[Alon2] N. Alon, Explicit Ramsey Graphs and Orthonormal Labelings, Electronic Journal of Combinatorics,
http://www.combinatorics.org/, #R12, 1 (1994), 8 pages.
[AlKS] N. Alon, M. Krivelevich and B. Sudakov, Tura´n Numbers of Bipartite Graphs and Related Ramsey
Type Questions, Combinatorics, Probability and Computing, 12 (2003) 477494.
[AlPu] N. Alon and P. Pudla´k, Constructive Lower Bounds for offdiagonal Ramsey Numbers, Israel Journal
of Mathematics, 122 (2001) 243251.
[AlRo¨] N. Alon and V. Ro¨dl, Asymptotically Tight Bounds for Some Multicored Ramsey Numbers, to
appear.
[AKM] J. Arste, K. Klamroth and I. Mengersen, Three Color Ramsey Numbers for Small Graphs, Utilitas
Mathematica, 49 (1996) 8596.
[AFM] M. Axenovich, Z. Fu¨redi and D. Mubayi, On Generalized Ramsey Theory: the Bipartite Case, Journal
of Combinatorial Theory, Series B, 79 (2000) 6686.
[BaRT]* A. Babak, S.P. Radziszowski and KungKuen Tse, Computation of the Ramsey Number R (B3,K5),
Bulletin of the Institute of Combinatorics and its Applications, 41 (2004) 7176.
[BaLS] P.N. Balister, J. Lehel and R.H. Schelp, Ramsey Unsaturated and Saturated Graphs, manuscript,
(2004).
[BaSS] P.N. Balister, R.H. Schelp and M. Simonovits, A Note on Ramsey SizeLinear Graphs, Journal of
Graph Theory, 39 (2002) 15.
[Bas] E.T. Baskoro, The Ramsey Number of Paths and Small Wheels, Majalah Ilmiah Himpunan Matematika
Indonesia, MIHMI, 8 (2002) 1316.
[BaSu] E.T. Baskoro and Surahmat, The Ramsey Number of Paths with respect to Wheels, preprint, (2002).
[BSNM] E.T. Baskoro, Surahmat, S.M. Nababan and M. Miller, On Ramsey Graph Numbers for Trees versus
Wheels of Five or Six Vertices, Graphs and Combinatorics, 18 (2002) 717721.
[] E.T. Baskoro, see also [SuBa1, SuBa2, SuBB1, SuBB2].
[Bev] D. Bevan, personal communication (2002).
[BS] A. Bialostocki and J. Scho¨nheim, On Some Tura´n and Ramsey Numbers for C4, in Graph Theory
and Combinatorics (ed. B. Bolloba´s), Academic Press, London, (1984) 2933.
[Bier] J. Bierbrauer, Ramsey Numbers for the Path with Three Edges, European Journal of Combinatorics,
7 (1986) 205206.
[BB] J. Bierbrauer and A. Brandis, On Generalized Ramsey Numbers for Trees, Combinatorica, 5 (1985)
95107.
[BJYHRZ] B. Bolloba´s, C.J. Jayawardene, Yang Jian Sheng, Huang Yi Ru, C.C. Rousseau, and Zhang Ke Min,
On a Conjecture Involving CycleComplete Graph Ramsey Numbers, Australasian Journal of Combinatorics,
22 (2000) 6371.
[BH] R. Bolze and H. Harborth, The Ramsey Number r (K4  x ,K5), in The Theory and Applications of
Graphs, (Kalamazoo, MI, 1980), John Wiley & Sons, New York, (1981) 109116.
[BoEr] J.A. Bondy and P. Erdo¨s, Ramsey Numbers for Cycles in Graphs, Journal of Combinatorial Theory,
Series B, 14 (1973) 4654.
[] A. Brandis, see [BB].
[Bra1] S. Brandt, Subtrees and Subforests in Graphs, Journal of Combinatorial Theory, Series B, 61 (1994)
6370.
[Bra2] S. Brandt, Sufficient Conditions for Graphs to Contain All Subgraphs of a Given Type, Ph.D. thesis,
Freie Universita¨t Berlin, 1994.
[Bra3] S. Brandt, Expanding Graphs and Ramsey Numbers, manuscript, (1996).
 29 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[BBH1]** S. Brandt, G. Brinkmann and T. Harmuth, All Ramsey Numbers r (K3,G) for Connected Graphs of
Order 9, Electronic Journal of Combinatorics, http://www.combinatorics.org/, #R7, 5 (1998), 20
pages.
[BBH2]** S. Brandt, G. Brinkmann and T. Harmuth, The Generation of Maximal TriangleFree Graphs, Graphs
and Combinatorics, 16 (2000) 149157.
[Brin]** G. Brinkmann, All Ramsey Numbers r (K3,G) for Connected Graphs of Order 7 and 8, Combinatorics,
Probability and Computing, 7 (1998) 129140.
[] G. Brinkmann, see also [BBH1, BBH2].
[] H.J. Broersma, see [SuBB1, SuBB2].
[BR]* J.P. Burling and S.W. Reyner, Some Lower Bounds of the Ramsey Numbers n (k , k ), Journal of
Combinatorial Theory, Series B, 13 (1972) 168169.
[Bu1] S.A. Burr, Generalized Ramsey Theory for Graphs  a Survey, in Graphs and Combinatorics (R. Bari
and F. Harary eds.), Springer LNM 406, Berlin, (1974) 5275.
[Bu2] S.A. Burr, Ramsey Numbers Involving Graphs with Long Suspended Paths, Journal of the London
Mathematical Society (2), 24 (1981) 405413.
[Bu3] S.A. Burr, Multicolor Ramsey Numbers Involving Graphs with Long Suspended Path, Discrete
Mathematics, 40 (1982) 1120.
[Bu4] S.A. Burr, Diagonal Ramsey Numbers for Small Graphs, Journal of Graph Theory, 7 (1983) 5769.
[Bu5] S.A. Burr, Ramsey Numbers Involving Powers, Ars Combinatoria, 15 (1983) 163168.
[Bu6] S.A. Burr, Determining Generalized Ramsey Numbers is NPHard, Ars Combinatoria, 17 (1984) 21
25.
[Bu7] S.A. Burr, What Can We Hope to Accomplish in Generalized Ramsey Theory?, Discrete Mathematics,
67 (1987) 215225.
[Bu8] S.A. Burr, On the Ramsey Numbers r (G, nH ) and r (nG , nH ) When n Is Large, Discrete Mathematics,
65 (1987) 215229.
[Bu9] S.A. Burr, On Ramsey Numbers for Large Disjoint Unions of Graphs, Discrete Mathematics, 70
(1988) 277293.
[BE1] S.A. Burr and P. Erdo¨s, Extremal Ramsey Theory for Graphs, Utilitas Mathematica, 9 (1976) 247
258.
[BE2] S.A. Burr and P. Erdo¨s, Generalizations of a RamseyTheoretic Result of Chva´tal, Journal of Graph
Theory, 7 (1983) 3951.
[BEFRS1] S.A. Burr, P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, An Extremal Problem in Generalized
Ramsey Theory, Ars Combinatoria, 10 (1980) 193203.
[BEFRS2] S.A. Burr, P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Ramsey Numbers for the Pair
Sparse GraphPath or Cycle, Transactions of the American Mathematical Society, 269 (1982) 501
512.
[BEFRS3] S.A. Burr, P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, On Ramsey Numbers Involving
Starlike Multipartite Graphs, Journal of Graph Theory, 7 (1983) 395409.
[BEFRS4] S.A. Burr, P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, The Ramsey Number for the Pair
Complete Bipartite GraphGraph of Limited Degree, in Graph Theory with Applications to Algorithms
and Computer Science, (Y. Alavi et al. eds.), John Wiley & Sons, New York, (1985) 163174.
[BEFRS5] S.A. Burr, P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Some Complete Bipartite Graph
Tree Ramsey Numbers, Annals of Discrete Mathematics, 41 (1989) 7989.
[BEFRSGJ]S.A. Burr, P. Erdo¨s, R.J. Faudree, C.C. Rousseau, R.H. Schelp, R.J. Gould and M.S. Jacobson, Goodness
of Trees for Generalized Books, Graphs and Combinatorics, 3 (1987) 16.
[BEFS] S.A. Burr, P. Erdo¨s, R.J. Faudree and R.H. Schelp, On the Difference between Consecutive Ramsey
Numbers, Utilitas Mathematica, 35 (1989) 115118.
 30 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[BES] S.A. Burr, P. Erdo¨s and J.H. Spencer, Ramsey Theorems for Multiple Copies of Graphs, Transactions
of the American Mathematical Society, 209 (1975) 8799.
[BF] S.A. Burr and R.J. Faudree, On Graphs G for Which All Large Trees Are Ggood, Graphs and Combinatorics,
9 (1993) 305313.
[BG] S.A. Burr and J.W. Grossman, Ramsey Numbers of Graphs with Long Tails, Discrete Mathematics,
41 (1982) 223227.
[BuRo1] S.A. Burr and J.A. Roberts, On Ramsey Numbers for Stars, Utilitas Mathematica, 4 (1973) 217220.
[BuRo2] S.A. Burr and J.A. Roberts, On Ramsey Numbers for Linear Forests, Discrete Mathematics, 8 (1974)
245250.
[Bush] L.E. Bush, The William Lowell Putnam Mathematical Competition, American Mathematical Monthly,
60 (1953) 539542.
[CET]* N.J. Calkin, P. Erdo¨s and C.A. Tovey, New Ramsey Bounds from Cyclic Graphs of Prime Order,
SIAM Journal of Discrete Mathematics, 10 (1997) 381387.
[Caro] Y. Caro, ZeroSum Problems  A Survey, Discrete Mathematics, 152 (1996) 93113.
[CLRZ] Y. Caro, Li Yusheng, C.C. Rousseau and Zhang Yuming, Asymptotic Bounds for Some Bipartite
Graph  Complete Graph Ramsey Numbers, Discrete Mathematics, 220 (2000) 5156.
[CGP] G. Chartrand, R.J. Gould and A.D. Polimeni, On Ramsey Numbers of Forests versus Nearly Complete
Graphs, Journal of Graph Theory, 4 (1980) 233239.
[CRSPS] G. Chartrand, C.C. Rousseau, M.J. Stewart, A.D. Polimeni and J. Sheehan, On StarBook Ramsey
Numbers, in Proceedings of the Fourth International Conference on the Theory and Applications of
Graphs, (Kalamazoo, MI 1980), John Wiley & Sons, (1981) 203214.
[CS] G. Chartrand and S. Schuster, On the existence of specified cycles in complementary graphs, Bulletin
of the American Mathematical Society, 77 (1971) 995998.
[Chen] Chen Guantao, A Result on C4Star Ramsey Numbers, Discrete Mathematics, 163 (1997) 243246.
[ChenS] Chen Guantao and R.H. Schelp, Graphs with Linearly Bounded Ramsey Numbers, Journal of Combinatorial
Theory, Series B, 57 (1993) 138149.
[ChenJ] Chen Jie, The Lower Bound of Some Ramsey Numbers (in Chinese), Journal of the Liaoning Normal
University, Natural Science, 25 (2002) 244246.
[ChenZZ1] Chen Yaojun, Zhang Yunqing and Zhang Ke Min, The Ramsey Numbers of Paths versus Wheels,
preprint, (2002).
[ChenZZ2] Chen Yaojun, Zhang Yunqing and Zhang Ke Min, The Ramsey Numbers of Stars versus Wheels, to
appear in the European Journal of Combinatorics, (2004).
[ChenZZ3] Chen Yaojun, Zhang Yunqing and Zhang Ke Min, The Ramsey Number R (Tn ,W6 ) for
D(Tn ) ³ n  3, Applied Mathematics Letters, 17 (2004) 281285.
[Chu1] F.R.K. Chung, On the Ramsey Numbers N(3,3,...,3 ; 2), Discrete Mathematics, 5 (1973) 317321.
[Chu2] F.R.K. Chung, On Triangular and Cyclic Ramsey Numbers with k Colors, in Graphs and Combinatorics
(R. Bari and F. Harary eds.), Springer LNM 406, Berlin, (1974) 236241.
[Chu3] F.R.K. Chung, A Note on Constructive Methods for Ramsey Numbers, Journal of Graph Theory, 5
(1981) 109113.
[Chu4] F.R.K. Chung, Open problems of Paul Erdo¨s in Graph Theory, Journal of Graph Theory, 25 (1997)
336.
[CCD] F.R.K. Chung, R. Cleve and P. Dagum, A Note on Constructive Lower Bounds for the Ramsey
Numbers R (3, t ), Journal of Combinatorial Theory, Series B, 57 (1993) 150155.
[ChGra1] F.R.K. Chung and R.L. Graham, On Multicolor Ramsey Numbers for Complete Bipartite Graphs,
Journal of Combinatorial Theory, Series B, 18 (1975) 164169.
[ChGra2] F.R.K. Chung and R.L. Graham, Erdo¨s on Graphs, His Legacy of Unsolved Problems, A K Peters,
Wellesley, Massachusetts (1998).
 31 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[ChGri] F.R.K. Chung and C.M. Grinstead, A Survey of Bounds for Classical Ramsey Numbers, Journal of
Graph Theory, 7 (1983) 2537.
[Chv] V. Chva´tal, TreeComplete Graph Ramsey Numbers, Journal of Graph Theory, 1 (1977) 93.
[CH1] V. Chva´tal and F. Harary, Generalized Ramsey Theory for Graphs, II. Small Diagonal Numbers,
Proceedings of the American Mathematical Society, 32 (1972) 389394.
[CH2] V. Chva´tal and F. Harary, Generalized Ramsey Theory for Graphs, III. Small OffDiagonal
Numbers, Pacific Journal of Mathematics, 41 (1972) 335345.
[CH3] V. Chva´tal and F. Harary, Generalized Ramsey Theory for Graphs, I. Diagonal Numbers, Periodica
Mathematica Hungarica, 3 (1973) 115124.
[CRST] V. Chva´tal, V. Ro¨dl, E. Szemere´di and W.T. Trotter Jr., The Ramsey Number of a Graph with
Bounded Maximum Degree, Journal of Combinatorial Theory, Series B, 34 (1983) 239243.
[ChvS] V. Chva´tal and A. Schwenk, On the Ramsey Number of the FiveSpoked Wheel, in Graphs and
Combinatorics (R. Bari and F. Harary eds.), Springer LNM 406, Berlin, (1974) 247261.
[Clan] M. Clancy, Some Small Ramsey Numbers, Journal of Graph Theory, 1 (1977) 8991.
[Clap] C. Clapham, The Ramsey Number r (C4,C4,C4), Periodica Mathematica Hungarica, 18 (1987) 317
318.
[CEHMS] C. Clapham, G. Exoo, H. Harborth, I. Mengersen and J. Sheehan, The Ramsey Number of K5  e ,
Journal of Graph Theory, 13 (1989) 715.
[Clark] L. Clark, On CycleStar Graph Ramsey Numbers, Congressus Numerantium, 50 (1985) 187192.
[] R. Cleve, see [CCD].
[Coc] E.J. Cockayne, Some TreeStar Ramsey Numbers, Journal of Combinatorial Theory, Series B, 17
(1974) 183187.
[CL1] E.J. Cockayne and P.J. Lorimer, The Ramsey Number for Stripes, Journal of the Australian
Mathematical Society, Series A, 19 (1975) 252256.
[CL2] E.J. Cockayne and P.J. Lorimer, On Ramsey Graph Numbers for Stars and Stripes, Canadian
Mathematical Bulletin, 18 (1975) 3134.
[CPR] B. Codenotti, P. Pudla´k and G. Resta, Some Structural Properties of LowRank Matrices Related to
Computational Complexity, Theoretical Computer Science, 235 (2000) 89107.
[CoPC] Special issue on Ramsey theory of Combinatorics, Probability and Computing, 12 (2003), Numbers 5
and 6.
[CsKo] R. Csa´ka´ny and J. Komlo´s, The Smallest Ramsey Numbers, Discrete Mathematics, 199 (1999) 193
199.
[] P. Dagum, see [CCD].
[Den] T. Denley, The Ramsey Numbers for Disjoint Unions of Cycles, Discrete Mathematics, 149 (1996)
3144.
[DuHu] Duan Chanlun and Huang Wenke, Lower Bound of Ramsey Number r (3, 10) (in Chinese), Acta
Scientiarum Naturalium Universitatis NeiMongol, 31 (2000) 468470.
[DLR] D. Duffus, H. Lefmann and V. Ro¨dl, Shift Graphs and Lower Bounds on Ramsey Numbers rk (l ; r ),
Discrete Mathematics, 137 (1995) 177187.
[DzKu] T. Dzido and M. Kubale, On Some Ramsey and Tura´ntype Numbers for Paths and Cycles,
manuscript, (2004).
[Ea1] Easy to obtain by simple combinatorics from other results, in particular by using graphs establishing
lower bounds with smaller parameters.
[Ea2] Unique 2(6,3,2) design gives lower bound 7, upper bound is easy.
[Eaton] N. Eaton, Ramsey Numbers for Sparse Graphs, Discrete Mathematics, 185 (1998) 6375.
[Eng] A. Engstro¨m, A Note on Two Multicolor Ramsey Numbers, manuscript, (2003).
 32 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[Erd1] P. Erdo¨s, Some Remarks on the Theory of Graphs, Bulletin of the American Mathematical Society,
53 (1947) 292294.
[Erd2] P. Erdo¨s, On the Combinatorial Problems Which I Would Most Like to See Solved, Combinatorica,
1 (1981) 2542.
[EFRS1] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Generalized Ramsey Theory for Multiple
Colors, Journal of Combinatorial Theory, Series B, 20 (1976) 250264.
[EFRS2] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, On CycleComplete Graph Ramsey
Numbers, Journal of Graph Theory, 2 (1978) 5364.
[EFRS3] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Ramsey Numbers for Brooms, Congressus
Numerantium, 35 (1982) 283293.
[EFRS4] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Multipartite GraphSparse Graph Ramsey
Numbers, Combinatorica, 5 (1985) 311318.
[EFRS5] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, A Ramsey Problem of Harary on Graphs
with Prescribed Size, Discrete Mathematics, 67 (1987) 227233.
[EFRS6] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Extremal Theory and Bipartite GraphTree
Ramsey Numbers, Discrete Mathematics, 72 (1988) 103112.
[EFRS7] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, The BookTree Ramsey Numbers, Scientia,
Series A: Mathematical Sciences, Valparaı´so, Chile, 1 (1988) 111117.
[EFRS8] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Multipartite GraphTree Graph Ramsey
Numbers, in Graph Theory and Its Applications: East and West, Proceedings of the First ChinaUSA
International Graph Theory Conference, Annals of the New York Academy of Sciences, 576 (1989)
146154.
[EFRS9] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Ramsey Size Linear Graphs, Combinatorics,
Probability and Computing, 2 (1993) 389399.
[EG] P. Erdo¨s and R.L. Graham, On Partition Theorems for Finite Sets, in Infinite and Finite Sets, (A.
Hajnal, R. Rado and V.T. So´s eds.) North Holland, (1975) 515527.
[] P. Erdo¨s, see also [BoEr, BE1, BE2, BEFRS1, BEFRS2, BEFRS3, BEFRS4, BEFRS5, BEFRSGJ,
BEFS, BES, CET].
[Ex1]* G. Exoo, Ramsey Numbers of Hypergraphs, Journal of Combinatorial Mathematics and Combinatorial
Computing, 2 (1987) 511.
[Ex2]* G. Exoo, Constructing Ramsey Graphs with a Computer, Congressus Numerantium, 59 (1987) 3136.
[Ex3]* G. Exoo, Applying Optimization Algorithm to Ramsey Problems, in Graph Theory, Combinatorics,
Algorithms, and Applications (Y. Alavi ed.), SIAM Philadelphia, (1989) 175179.
[Ex4]* G. Exoo, A Lower Bound for R (5, 5), Journal of Graph Theory, 13 (1989) 9798.
[Ex5]* G. Exoo, On Two Classical Ramsey Numbers of the Form R (3, n ), SIAM Journal of Discrete
Mathematics, 2 (1989) 488490.
[Ex6]* G. Exoo, A Lower Bound for r (K5  e ,K5), Utilitas Mathematica, 38 (1990) 187188.
[Ex7]* G. Exoo, On the Three Color Ramsey Number of K4  e , Discrete Mathematics, 89 (1991) 301305.
[Ex8]* G. Exoo, Indiana State University, personal communication (1992).
[Ex9]* G. Exoo, Announcement: On the Ramsey Numbers R (4, 6), R (5, 6) and R (3,12), Ars Combinatoria,
35 (1993) 85. The construction of a graph proving R (4, 6) ³ 35 is presented in detail at
http://ginger.indstate.edu/ge/RAMSEY (2001).
[Ex10]* G. Exoo, A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers of K3, Electronic
Journal of Combinatorics, http://www.combinatorics.org/, #R8, 1 (1994), 3 pages.
[Ex11]* G. Exoo, Indiana State University, personal communication (1997).
[Ex12]* G. Exoo, Some New Ramsey Colorings, Electronic Journal of Combinatorics,
http://www.combinatorics.org/, #R29, 5 (1998), 5 pages. The constructions are available electronically
 33 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
from http://ginger.indstate.edu/ge/RAMSEY.
[Ex13]* G. Exoo, Indiana State University, personal communication (1998). Constructions available at
http://ginger.indstate.edu/ge/RAMSEY.
[Ex14]* G. Exoo, Indiana State University, New Lower Bounds for Table III, (2000). Constructions available
at http://ginger.indstate.edu/ge/RAMSEY.
[Ex15]* G. Exoo, Some Applications of pq groups in Graph Theory, Discussiones Mathematicae Graph
Theory, 24 (2004) 109114. Constructions available at http://ginger.indstate.edu/ge/RAMSEY.
[Ex16]* G. Exoo, Indiana State University, personal communication (20022004). Constructions available at
http://ginger.indstate.edu/ge/RAMSEY.
[EHM1] G. Exoo, H. Harborth and I. Mengersen, The Ramsey Number of K4 versus K5  e , Ars Combinatoria,
25A (1988) 277286.
[EHM2] G. Exoo, H. Harborth and I. Mengersen, On Ramsey Number of K2,n , in Graph Theory, Combinatorics,
Algorithms, and Applications (Y. Alavi, F.R.K. Chung, R.L. Graham and D.F. Hsu eds.), SIAM
Philadelphia, (1989) 207211.
[ExRe]* G. Exoo and D.F. Reynolds, Ramsey Numbers Based on C5Decompositions, Discrete Mathematics,
71 (1988) 119127.
[] G. Exoo, see also [CEHMS, XXER].
[FLPS] R.J. Faudree, S.L. Lawrence, T.D. Parsons and R.H. Schelp, PathCycle Ramsey Numbers, Discrete
Mathematics, 10 (1974) 269277.
[FM]** R.J. Faudree and B.D. McKay, A Conjecture of Erdo¨s and the Ramsey Number r (W6), Journal of
Combinatorial Mathematics and Combinatorial Computing, 13 (1993) 2331.
[FRS1] R.J. Faudree, C.C. Rousseau and R.H. Schelp, All TriangleGraph Ramsey Numbers for Connected
Graphs of Order Six, Journal of Graph Theory, 4 (1980) 293300.
[FRS2] R.J. Faudree, C.C. Rousseau and R.H. Schelp, Studies Related to the Ramsey Number r (K5  e ), in
Graph Theory and Its Applications to Algorithms and Computer Science, (Y. Alavi et al. eds.), John
Wiley and Sons, New York, (1985) 251271.
[FRS3] R.J. Faudree, C.C. Rousseau and R.H. Schelp, Generalizations of the TreeComplete Graph Ramsey
Number, in Graphs and Applications, (F. Harary and J.S. Maybee eds.), John Wiley and Sons, New
York, (1985) 117126.
[FRS4] R.J. Faudree, C.C. Rousseau and R.H. Schelp, Small Order GraphTree Ramsey Numbers, Discrete
Mathematics, 72 (1988) 119127.
[FRS5] R.J. Faudree, C.C. Rousseau and R.H. Schelp, A Good Idea in Ramsey Theory, in Graph Theory,
Combinatorics, Algorithms, and Applications (San Francisco, CA 1989), SIAM Philadelphia, PA
(1991) 180189.
[FRS6] R.J. Faudree, C.C. Rousseau and J. Sheehan, More from the Good Book, in Proceedings of the Ninth
Southeastern Conference on Combinatorics, Graph Theory, and Computing, Utilitas Mathematica
Publ., Congressus Numerantium XXI (1978) 289299.
[FRS7] R.J. Faudree, C.C. Rousseau and J. Sheehan, Strongly Regular Graphs and Finite Ramsey Theory,
Linear Algebra and its Applications, 46 (1982) 221241.
[FRS8] R.J. Faudree, C.C. Rousseau and J. Sheehan, CycleBook Ramsey Numbers, Ars Combinatoria, 31
(1991) 239248.
[FS1] R.J. Faudree and R.H. Schelp, All Ramsey Numbers for Cycles in Graphs, Discrete Mathematics, 8
(1974) 313329.
[FS2] R.J. Faudree and R.H. Schelp, Path Ramsey Numbers in Multicolorings, Journal of Combinatorial
Theory, Series B, 19 (1975) 150160.
[FS3] R.J. Faudree and R.H. Schelp, Ramsey Numbers for All Linear Forests, Discrete Mathematics, 16
(1976) 149155.
 34 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[FS4] R.J. Faudree and R.H. Schelp, Some Problems in Ramsey Theory, in Theory and Applications of
Graphs, (conference proceedings, Kalamazoo, MI 1976), Lecture Notes in Mathematics 642,
Springer, Berlin, (1978) 500515.
[FSR] R.J. Faudree, R.H. Schelp and C.C. Rousseau, Generalizations of a Ramsey Result of Chva´tal, in
Proceedings of the Fourth International Conference on the Theory and Applications of Graphs,
(Kalamazoo, MI 1980), John Wiley & Sons, (1981) 351361.
[FSS1] R.J. Faudree, R.H. Schelp and M. Simonovits, On Some Ramsey Type Problems Connected with
Paths, Cycles and Trees, Ars Combinatoria, 29A (1990) 97106.
[FSS2] R.J. Faudree, A. Schelten and I. Schiermeyer, The Ramsey Number r (C7,C7,C7), Discussiones
Mathematicae Graph Theory, 23 (2003) 141158.
[FS] R.J. Faudree and M. Simonovits, Ramsey Problems and Their Connection to Tura´nType Extremal
Problems, Journal of Graph Theory, 16 (1992) 2550.
[] R.J. Faudree, see also [BEFRS1, BEFRS2, BEFRS3, BEFRS4, BEFRS5, BEFRSGJ, BEFS, BF,
EFRS1, EFRS2, EFRS3, EFRS4, EFRS5, EFRS6, EFRS7, EFRS8, EFRS9].
[FKR]** S. Fettes, R.L. Kramer and S.P. Radziszowski, An Upper Bound of 62 on the Classical Ramsey
Number R (3,3,3,3), to appear in Ars Combinatoria, (2004).
[Fo] J. Folkman, Notes on the Ramsey Number N(3,3,3,3), Journal of Combinatorial Theory, Series A, 16
(1974) 371379.
[FraWi] P. Frankl and R.M. Wilson, Intersection Theorems with Geometric Consequences, Combinatorica, 1
(1981) 357368.
[Fra1] K. Fraughnaugh Jones, Independence in Graphs with Maximum Degree Four, Journal of Combinatorial
Theory, Series B, 37 (1984) 254269.
[Fra2] K. Fraughnaugh Jones, Size and Independence in TriangleFree Graphs with Maximum Degree
Three, Journal of Graph Theory, 14 (1990) 525535.
[FrLo] K. Fraughnaugh and S.C. Locke, Finding Independent Sets in Trianglefree Graphs, SIAM Journal of
Discrete Mathematics, 9 (1996) 674681.
[Fre] H. Fredricksen, Schur Numbers and the Ramsey Numbers N(3,3,...,3 ; 2), Journal of Combinatorial
Theory, Series A, 27 (1979) 376377.
[FreSw]* H. Fredricksen and M.M. Sweet, Symmetric SumFree Partitions and Lower Bounds for Schur
Numbers, Electronic Journal of Combinatorics, http://www.combinatorics.org/, #R32, 7 (2000), 9
pages.
[] Z. Fu¨redi, see [AFM].
[GeGy] L. Gerencse´r and A. Gya´rfa´s, On RamseyType Problems, Annales Universitatis Scientiarum
Budapestinensis, Eo¨tvo¨s Sect. Math., 10 (1967) 167170.
[Gi1] G. Giraud, Une ge´ne´ralisation des nombres et de l’ine´galite´ de Schur, C.R. Acad. Sc. Paris, Se´ries AB,
266 (1968) A437A440.
[Gi2] G. Giraud, Minoration de certains nombres de Ramsey binaires par les nombres de Schur ge´ne´ralise´s,
C.R. Acad. Sc. Paris, Se´ries AB, 266 (1968) A481A483.
[Gi3] G. Giraud, Nouvelles majorations des nombres de Ramsey binairesbicolores, C.R. Acad. Sc. Paris,
Se´ries AB, 268 (1969) A5A7.
[Gi4] G. Giraud, Majoration du nombre de Ramsey ternairebicolore en (4,4), C.R. Acad. Sc. Paris, Se´ries
AB, 269 (1969) A620A622.
[Gi5] G. Giraud, Une minoration du nombre de quadrangles unicolores et son application a` la majoration
des nombres de Ramsey binairesbicolores, C.R. Acad. Sc. Paris, Se´ries AB, 276 (1973) A1173
A1175.
[Gi6] G. Giraud, Sur le proble`me de Goodman pour les quadrangles et la majoration des nombres de Ramsey,
Journal of Combinatorial Theory, Series B, 27 (1979) 237253.
 35 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[] A.M. Gleason, see [GG].
[GK] W. Goddard and D.J. Kleitman, An upper bound for the Ramsey numbers r (K3,G), Discrete
Mathematics, 125 (1994) 177182.
[GoMC] A. Gonc¸alves and E.L. Monte Carmelo, Some Geometric Structures and Bounds for Ramsey
Numbers, Discrete Mathematics, 280 (2004) 2938.
[GJ1] R.J. Gould and M.S. Jacobson, Bounds for the Ramsey Number of a Disconnected Graph Versus Any
Graph, Journal of Graph Theory, 6 (1982) 413417.
[GJ2] R.J. Gould and M.S. Jacobson, On the Ramsey Number of Trees Versus Graphs with Large Clique
Number, Journal of Graph Theory, 7 (1983) 7178.
[] R.J. Gould, see also [BEFRSGJ, CGP].
[GrRo¨] R.L. Graham and V. Ro¨dl, Numbers in Ramsey Theory, in Surveys in Combinatorics, (ed. C. Whitehead),
Cambridge University Press, 1987.
[GRR1] R.L. Graham, V. Ro¨dl and A. Rucin´ski, On Graphs with Linear Ramsey Numbers, Journal of Graph
Theory, 35 (2000) 176192.
[GRR2] R.L. Graham, V. Ro¨dl and A. Rucin´ski, On Bipartite Graphs with Linear Ramsey Numbers, Paul
Erdo¨s and his mathematics, Combinatorica, 21 (2001) 199209.
[GRS] R.L. Graham, B.L. Rothschild and J.H. Spencer, Ramsey Theory, John Wiley & Sons, 1990.
[] R.L. Graham, see also [ChGra1, ChGra2, EG].
[GY] J.E. Graver and J. Yackel, Some Graph Theoretic Results Associated with Ramsey’s Theorem, Journal
of Combinatorial Theory, 4 (1968) 125175.
[GG] R.E. Greenwood and A.M. Gleason, Combinatorial Relations and Chromatic Graphs, Canadian Journal
of Mathematics, 7 (1955) 17.
[GH] U. Grenda and H. Harborth, The Ramsey Number r (K3,K7  e ), Journal of Combinatorics, Information
& System Sciences, 7 (1982) 166169.
[Gri] J.R. Griggs, An Upper Bound on the Ramsey Numbers R (3,k ), Journal of Combinatorial Theory,
Series A, 35 (1983) 145153.
[GR]** C. Grinstead and S. Roberts, On the Ramsey Numbers R (3,8) and R (3,9), Journal of Combinatorial
Theory, Series B, 33 (1982) 2751.
[] C. Grinstead, see also [ChGri].
[Grol1] V. Grolmusz, Superpolynomial Size SetSystems with Restricted Intersections mod 6 and Explicit
Ramsey Graphs, Combinatorica, 20 (2000) 7388.
[Grol2] V. Grolmusz, Low Rank CoDiagonal Matrices and Ramsey Graphs, Electronic Journal of Combinatorics,
http://www.combinatorics.org/, #R15, 7 (2000) 7 pages.
[Grol3] V. Grolmusz, SetSystems with Restricted Multiple Intersections, Electronic Journal of Combinatorics,
http://www.combinatorics.org/, #R8, 9 (2002) 10 pages.
[Gros1] J.W. Grossman, Some Ramsey Numbers of Unicyclic Graphs, Ars Combinatoria, 8 (1979) 5963.
[Gros2] J.W. Grossman, The Ramsey Numbers of the Union of Two Stars, Utilitas Mathematica, 16 (1979)
271279.
[GHK] J.W. Grossman, F. Harary and M. Klawe, Generalized Ramsey Theory for Graphs, X: Double Stars,
Discrete Mathematics, 28 (1979) 247254.
[] J.W. Grossman, see also [BG].
[GV] Guo Yubao and L. Volkmann, TreeRamsey Numbers, Australasian Journal of Combinatorics, 11
(1995) 169175.
[] L. Gupta, see [GGS].
[GGS] S.K. Gupta, L. Gupta and A. Sudan, On Ramsey Numbers for FanFan Graphs, Journal of Combinatorics,
Information & System Sciences, 22 (1997) 8593.
 36 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[GT] A. Gya´rfa´s and Z. Tuza, An Upper Bound on the Ramsey Number of Trees, Discrete Mathematics,
66 (1987) 309310.
[] A. Gya´rfa´s, see also [GeGy].
[Haa]* H. Haanpa¨a¨, A Lower Bound for a Ramsey Number, Congressus Numerantium, 144 (2000) 189191.
[Ha¨g] R. Ha¨ggkvist, On the PathComplete Bipartite Ramsey Number, Discrete Mathematics, 75 (1989)
243245.
[Han]* D. Hanson, Sumfree Sets and Ramsey Numbers, Discrete Mathematics, 14 (1976) 5761.
[] D. Hanson, see also [AbbH].
[Har1] F. Harary, Recent Results on Generalized Ramsey Theory for Graphs, in Graph Theory and Applications,
(Y. Alavi et al. eds.) Springer, Berlin (1972) 125138.
[Har2] F. Harary, Generalized Ramsey Theory I to XIII: Achievement and Avoidance Numbers, in Proceedings
of the Fourth International Conference on the Theory and Applications of Graphs, (Kalamazoo,
MI 1980), John Wiley & Sons, (1981) 373390.
[] F. Harary, see also [CH1, CH2, CH3, GHK].
[HaKr]** H. Harborth and S. Krause, Ramsey Numbers for Circulant Colorings, Congressus Numerantium, 161
(2003) 139150.
[HaMe1] H. Harborth and I. Mengersen, An Upper Bound for the Ramsey Number r (K5  e ), Journal of
Graph Theory, 9 (1985) 483485.
[HaMe2] H. Harborth and I. Mengersen, All Ramsey Numbers for Five Vertices and Seven or Eight Edges,
Discrete Mathematics, 73 (1988/89) 9198.
[HaMe3] H. Harborth and I. Mengersen, The Ramsey Number of K3,3, in Combinatorics, Graph Theory, and
Applications, Vol. 2 (Y. Alavi, G. Chartrand, O.R. Oellermann and J. Schwenk eds.), John Wiley &
Sons, (1991) 639644.
[] H. Harborth, see also [BH, CEHMS, EHM1, EHM2, GH].
[HaMe4] M. Harborth and I. Mengersen, Some Ramsey Numbers for Complete Bipartite Graphs, Australasian
Journal of Combinatorics, 13 (1996) 119128.
[] T. Harmuth, see [BBH1, BBH2].
[HaŁT] P.E. Haxell, T. Łuczak and P.W. Tingley, Ramsey Numbers for Trees of Small Maximum Degree,
Combinatorica, 22 (2002) 287320.
[Hein] K. Heinrich, Proper Colourings of K15, Journal of the Australian Mathematical Society, Series A, 24
(1977) 465495.
[He1] G.R.T. Hendry, Diagonal Ramsey Numbers for Graphs with Seven Edges, Utilitas Mathematica, 32
(1987) 1134.
[He2] G.R.T. Hendry, Ramsey Numbers for Graphs with Five Vertices, Journal of Graph Theory, 13 (1989)
245248.
[He3] G.R.T. Hendry, The Ramsey Numbers r (K2 + K3,K4) and r (K1 + C4,K4), Utilitas Mathematica, 35
(1989) 4054, addendum in 36 (1989) 2532.
[He4] G.R.T. Hendry, Critical Colorings for Clancy’s Ramsey Numbers, Utilitas Mathematica, 41 (1992)
181203.
[He5] G.R.T. Hendry, Small Ramsey Numbers II. Critical Colorings for r (C5 + e ,K5), Quaestiones
Mathematica, 17 (1994) 249258.
[] G.R.T. Hendry, see also [YH].
[HiIr]* R. Hill and R.W. Irving, On Group Partitions Associated with Lower Bounds for Symmetric Ramsey
Numbers, European Journal of Combinatorics, 3 (1982) 3550.
[Hir] J. Hirschfeld, A Lower Bound for Ramsey’s Theorem, Discrete Mathematics, 32 (1980) 8991.
[HoMe] M. Hoeth and I. Mengersen, Ramsey Numbers for Graphs of Order Four versus Connected Graphs of
Order Six, Utilitas Mathematica, 57 (2000) 319.
 37 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[HuSo] Huang Da Ming and Song En Min, Properties and Lower Bounds of the Third Order Ramsey
Numbers (in Chinese), Mathematica Applicata, 9 (1996) 105107.
[Hua1] Huang Guotai, Some Generalized Ramsey Numbers (in Chinese), Mathematica Applicata, 1 (1988)
97101.
[Hua2] Huang Guotai, An Unsolved Problem of Gould and Jacobson (in Chinese), Mathematica Applicata, 9
(1996) 234236.
[] Huang Wenke, see [DuHu].
[HZ1] Huang Yi Ru and Zhang Ke Min, An New Upper Bound Formula for Two Color Classical Ramsey
Numbers, Journal of Combinatorial Mathematics and Combinatorial Computing, 28 (1998) 347350.
[HZ2] Huang Yi Ru and Zhang Ke Min, New Upper Bounds for Ramsey Numbers, European Journal of
Combinatorics, 19 (1998) 391394.
[] Huang Yi Ru, see also [BJYHRZ, YHZ1, YHZ2].
[Ir] R.W. Irving, Generalised Ramsey Numbers for Small Graphs, Discrete Mathematics, 9 (1974) 251
264.
[] R.W. Irving, see also [HiIr].
[Is1] J.R. Isbell, N(4,4 ; 3) ³ 13, Journal of Combinatorial Theory, 6 (1969) 210.
[Is2] J.R. Isbell, N(5,4 ; 3) ³ 24, Journal of Combinatorial Theory, Series A, 34 (1983) 379380.
[Jac] M.S. Jacobson, On the Ramsey Number for Stars and a Complete Graph, Ars Combinatoria, 17
(1984) 167172.
[] M.S. Jacobson, see also [BEFRSGJ, GJ1, GJ2].
[JR1] C.J. Jayawardene and C.C. Rousseau, An Upper Bound for the Ramsey Number of a Quadrilateral
versus a Complete Graph on Seven Vertices, Congressus Numerantium, 130 (1998) 175188.
[JR2] C.J. Jayawardene and C.C. Rousseau, Ramsey Numbers r (C6,G) for All Graphs G of Order Less
than Six, Congressus Numerantium, 136 (1999) 147159.
[JR3] C.J. Jayawardene and C.C. Rousseau, The Ramsey Numbers for a Quadrilateral vs. All Graphs on Six
Vertices, Journal of Combinatorial Mathematics and Combinatorial Computing, 35 (2000) 7187.
[JR4] C.J. Jayawardene and C.C. Rousseau, Ramsey Numbers r (C5,G) for All Graphs G of Order Six, Ars
Combinatoria, 57 (2000) 163173.
[JR5] C.J. Jayawardene and C.C. Rousseau, The Ramsey Number for a Cycle of Length Five vs. a Complete
Graph of Order Six, Journal of Graph Theory, 35 (2000) 99108.
[] C.J. Jayawardene, see also [BJYHRZ, RoJa1, RoJa2].
[Jin]** Jin Xia, Ramsey Numbers Involving a Triangle: Theory & Applications, Technical Report RITTR
93019, MS thesis, Department of Computer Science, Rochester Institute of Technology, 1993.
[] Jin Xia, see also [RaJi].
[JGT] Special volume on Ramsey theory of Journal of Graph Theory, Volume 7, Number 1, (1983).
[Ka1] J.G. Kalbfleisch, Construction of Special EdgeChromatic Graphs, Canadian Mathematical Bulletin, 8
(1965) 575584.
[Ka2]* J.G. Kalbfleisch, Chromatic Graphs and Ramsey’s Theorem, Ph.D. thesis, University of Waterloo,
January 1966.
[Ka3] J.G. Kalbfleisch, On Robillard’s Bounds for Ramsey Numbers, Canadian Mathematical Bulletin, 14
(1971) 437440.
[KaSt] J.G. Kalbfleisch and R.G. Stanton, On the Maximal TriangleFree EdgeChromatic Graphs in Three
Colors, Journal of Combinatorial Theory, 5 (1968) 920.
[Ka´Ros] G. Ka´rolyi and V. Rosta, Generalized and Geometric Ramsey Numbers for Cycles, Theoretical Computer
Science, 263 (2001) 8798.
[Ke´ry] G. Ke´ry, On a Theorem of Ramsey (in Hungarian), Matematikai Lapok, 15 (1964) 204224.
 38 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[Kim] J.H. Kim, The Ramsey Number R (3, t ) has Order of Magnitude t
2
/ log t , Random Structures and
Algorithms, 7 (1995) 173207.
[KM1] K. Klamroth and I. Mengersen, Ramsey Numbers of K3 versus (p , q )Graphs, Ars Combinatoria, 43
(1996) 107120.
[KM2] K. Klamroth and I. Mengersen, The Ramsey Number of r (K1,3,C4,K4 ), Utilitas Mathematica, 52
(1997) 6581.
[] K. Klamroth, see also [AKM].
[] M. Klawe, see [GHK].
[] D.J. Kleitman, see [GK].
[Ko¨h] W. Ko¨hler, On a Conjecture by Grossman, Ars Combinatoria, 23 (1987) 103106.
[] J. Komlo´s, see [CsKo, AKS].
[KoRo¨1] A.V. Kostochka and V. Ro¨dl, On Graphs with Small Ramsey Numbers, Journal of Graph Theory, 37
(2001) 198204.
[KoRo¨2] A.V. Kostochka and V. Ro¨dl, On Graphs with Small Ramsey Numbers, II, to appear in Combinatorica,
(2004).
[KoSu] A.V. Kostochka and B. Sudakov, On Ramsey Numbers of Sparse Graphs, Combinatorics, Probability
and Computing, 12 (2003) 627641.
[] R.L. Kramer, see [FKR].
[KrRod] I. Krasikov and Y. Roditty, On Some Ramsey Numbers of Unicyclic Graphs, Bulletin of the Institute
of Combinatorics and its Applications, 33 (2001) 2934.
[] S. Krause, see [HaKr].
[KLR]* D.L. Kreher, Li Wei and S.P. Radziszowski, Lower Bounds for MultiColored Ramsey Numbers
From Group Orbits, Journal of Combinatorial Mathematics and Combinatorial Computing, 4 (1988)
8795.
[] D.L. Kreher, see also [RK1, RK2, RK3, RK4].
[Kriv] M. Krivelevich, Bounding Ramsey Numbers through Large Deviation Inequalities, Random Structures
and Algorithms, 7 (1995) 145155.
[] M. Krivelevich, see also [AlKS].
[] M. Kubale, see [DzKu].
[] P.C.B. Lam, see [ShiuLL].
[La1] S.L. Lawrence, CycleStar Ramsey Numbers, Notices of the American Mathematical Society, 20
(1973) Abstract A 420.
[La2] S.L. Lawrence, Bipartite Ramsey Theory, Notices of the American Mathematical Society, 20 (1973)
Abstract A 562.
[] S.L. Lawrence, see also [FLPS].
[LayMa] C. Laywine and J.P. Mayberry, A Simple Construction Giving the Two Nonisomorphic Triangle
Free 3Colored K16’s, Journal of Combinatorial Theory, Series B, 45 (1988) 120124.
[LaMu] F. Lazebnik and D. Mubayi, New Lower Bounds for Ramsey Numbers of Graphs and Hypergraphs,
Advances in Applied Mathematics, 28 (2002) 544559.
[LaWo1] F. Lazebnik and A. Woldar, New Lower Bounds on the Multicolor Ramsey Numbers rk (C4 ), Journal
of Combinatorial Theory, Series B, 79 (2000) 172176.
[LaWo2] F. Lazebnik and A. Woldar, General Properties of Some Families of Graphs Defined by Systems of
Equations, Journal of Graph Theory, 38 (2001) 6586.
[Lef] H. Lefmann, Ramsey Numbers for Monotone Paths and Cycles, Ars Combinatoria, 35 (1993) 271
279.
 39 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[] H. Lefmann, see also [DLR].
[] J. Lehel, see [BaLS].
[Les]* A. Lesser, Theoretical and Computational Aspects of Ramsey Theory, Examensarbeten i Matematik,
Matematiska Institutionen, Stockholms Universitet, 3 (2001).
[LiWa1] Li Da Yong and Wang Zhi Jian, The Ramsey Number r (mC4, nC4 ) (in Chinese), Journal of
Shanghai Tiedao University, 20 (1999) 6670, 83.
[Liwa2] Li Da Yong and Wang Zhi Jian, The Ramsey Numbers r (mC4, nC5 ), Journal of Combinatorial
Mathematics and Combinatorial Computing, 45 (2003) 245252.
[] Li Guiqing, see [LSZL, SLLL, SLL1, SLZL].
[] Li Jinwen, see [ZLLS].
[] Li Qiao, see [SLLL, SLL2, SLL3].
[] Li Wei, see [KLR].
[Li] Li Yusheng, Some Ramsey Numbers of Graphs with Bridge, Journal of Combinatorial Mathematics
and Combinatorial Computing, 25 (1997) 225229.
[LR1] Li Yusheng and C.C. Rousseau, On BookComplete Graph Ramsey Numbers, Journal of Combinatorial
Theory, Series B, 68 (1996) 3644.
[LR2] Li Yusheng and C.C. Rousseau, FanComplete Graph Ramsey Numbers, Journal of Graph Theory, 23
(1996) 413420.
[LR3] Li Yusheng and C.C. Rousseau, On the Ramsey Number r (H + Kn ,Kn ), Discrete Mathematics, 170
(1997) 265267.
[LR4] Li Yusheng and C.C. Rousseau, A Ramsey Goodness Result for Graphs with Many Pendant Edges,
Ars Combinatoria, 49 (1998) 315318.
[LRS] Li Yusheng, C.C. Rousseau and L. Solte´s, Ramsey Linear Families and Generalized Subdivided
Graphs, Discrete Mathematics, 170 (1997) 269275.
[LRZ] Li Yusheng, C.C. Rousseau and Zang Wenan, Asymptotic Upper Bounds for Ramsey Functions,
Graphs and Combinatorics, 17 (2001) 123128.
[LZ] Li Yusheng and Zang Wenan, Ramsey Numbers Involving Large Dense Graphs and Bipartite Tura´n
Numbers, Journal of Combinatorial Theory, Series B, 87 (2003) 280288.
[] Li Yusheng, see also [CLRZ, ShiuLL, ZaLi].
[] Li Zhenchong, see [LSL].
[Lind] B. Lindstro¨m, Undecided RamseyNumbers for Paths, Discrete Mathematics, 43 (1983) 111112.
[Ling] A.C.H. Ling, Some Applications of Combinatorial Designs to Extremal Graph Theory, Ars Combinatoria,
67 (2003) 221229.
[] Andy Liu, see [AbbL].
[] Liu Linzhong, see [ZLLS].
[] Liu Yanwu, see [SYL].
[Loc] S.C. Locke, Bipartite Density and the Independence Ratio, Journal of Graph Theory, 10 (1986) 47
53.
[] S.C. Locke, see also [FrLo].
[Lor] P.J. Lorimer, The Ramsey Numbers for Stripes and One Complete Graph, Journal of Graph Theory,
8 (1984) 177184.
[LorMu] P.J. Lorimer and P.R. Mullins, Ramsey Numbers for Quadrangles and Triangles, Journal of Combinatorial
Theory, Series B, 23 (1977) 262265.
[LorSe] P.J. Lorimer and R.J. Segedin, Ramsey Numbers for Multiple Copies of Complete Graphs, Journal of
Graph Theory, 2 (1978) 8991.
 40 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[LorSo] P.J. Lorimer and W. Solomon, The Ramsey Numbers for Stripes and Complete Graphs 1, Discrete
Mathematics, 104 (1992) 9197. Corrigendum in Discrete Mathematics, 131 (1994) 395.
[] P.J. Lorimer, see also [CL1, CL2].
[LorMe1] R. Lortz and I. Mengersen, On the Ramsey Numbers r (K2,n  1,K2,n ) and r (K2,n ,K2,n ), Utilitas
Mathematica, 61 (2002) 8795.
[LorMe2] R. Lortz and I. Mengersen, Bounds on Ramsey Numbers of Certain Complete Bipartite Graphs,
Results in Mathematics, 41 (2002) 140149.
[LorMe3]* R. Lortz and I. Mengersen, OffDiagonal and Asymptotic Results on the Ramsey Number
r (K2,m ,K2,n ), Journal of Graph Theory, 43 (2003) 252268.
[Łuc] T. Łuczak, R (Cn ,Cn ,Cn ) £ (4 + o (1)) n , Journal of Combinatorial Theory, Series B, 75 (1999) 174
187.
[] T. Łuczak, see also [HaŁT].
[LSL]* Luo Haipeng, Su Wenlong and Li Zhenchong, The Properties of SelfComplementary Graphs and
New Lower Bounds for Diagonal Ramsey Numbers, Australasian Journal of Combinatorics, 25
(2002) 103116.
[LSS]* Luo Haipeng, Su Wenlong and Shen YunQiu, New Lower Bounds of Ten Classical Ramsey
Numbers, Australasian Journal of Combinatorics, 24 (2001) 8190.
[LSZL]* Luo Haipeng, Su Wenlong, Zhang Zhengyou and Li Guiqing, New Lower Bounds for Twelve Classical
2Color Ramsey Numbers R (k , l ) (in Chinese), Guangxi Sciences, 7, 2 (2000) 120121.
[] Luo Haipeng, see also [SL, SLLL, SLL1, SLL2, SLL3, SLZL, ZSL].
[Mac]* J. Mackey, Combinatorial Remedies, Ph.D. Thesis, Department of Mathematics, University of
Hawaii, 1994.
[Mat]* R. Mathon, Lower Bounds for Ramsey Numbers and Association Schemes, Journal of Combinatorial
Theory, Series B, 42 (1987) 122127.
[] J.P. Mayberry, see [LayMa].
[McS] C. McDiarmid and A. Steger, Tidier Examples for Lower Bounds on Diagonal Ramsey Numbers,
Journal of Combinatorial Theory, Series A, 74 (1996) 147152.
[McK]** B.D. McKay, Australian National University, personal communication (2003). Graphs available at
http://cs.anu.edu.au/people/bdm/data/ramsey.html.
[MPR]** B.D. McKay, K. Piwakowski and S.P. Radziszowski, Ramsey Numbers for Triangles versus Almost
Complete Graphs, to appear in Ars Combinatoria, (2004).
[MR1]** B.D. McKay and S.P. Radziszowski, The First Classical Ramsey Number for Hypergraphs is Computed,
Proceedings of the Second Annual ACMSIAM Symposium on Discrete Algorithms, SODA’91,
San Francisco, (1991) 304308.
[MR2]* B.D. McKay and S.P. Radziszowski, A New Upper Bound for the Ramsey Number R (5, 5), Australasian
Journal of Combinatorics, 5 (1992) 1320.
[MR3]** B.D. McKay and S.P. Radziszowski, Linear Programming in Some Ramsey Problems, Journal of
Combinatorial Theory, Series B, 61 (1994) 125132.
[MR4]** B.D. McKay and S.P. Radziszowski, R (4, 5) = 25, Journal of Graph Theory, 19 (1995) 309322.
[MR5]** B.D. McKay and S.P. Radziszowski, Subgraph Counting Identities and Ramsey Numbers, Journal of
Combinatorial Theory, Series B, 69 (1997) 193209.
[MZ]** B.D. McKay and Zhang Ke Min, The Value of the Ramsey Number R (3,8), Journal of Graph
Theory, 16 (1992) 99105.
[] B.D. McKay, see also [FM].
[McN]** J. McNamara, SUNY Brockport, personal communication (1995).
[McR]** J. McNamara and S.P. Radziszowski, The Ramsey Numbers R (K4  e ,K6  e ) and R (K4  e ,K7  e ),
Congressus Numerantium, 81 (1991) 8996.
 41 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[MO] I. Mengersen and J. Oeckermann, MatchingStar Ramsey Sets, Discrete Applied Mathematics, 95
(1999) 417424.
[] I. Mengersen, see also [AKM, CEHMS, EHM1, EHM2, HoMe, HaMe1, HaMe2, HaMe3, HaMe4,
KM1, KM2, LorMe1, LorMe2, LorMe3].
[] M. Miller, see [BSNM].
[MS] H. Mizuno and I. Sato, Ramsey Numbers for Unions of Some Cycles, Discrete Mathematics, 69
(1988) 283294.
[] E.L. Monte Carmelo, see [GoMC].
[] D. Mubayi, see [AFM, LaMu].
[] P.R. Mullins, see [LorMu].
[] S.M. Nababan, see [BSNM].
[Nes˘] J. Nes˘etr˘il, Ramsey Theory, chapter 25 in Handbook of Combinatorics, ed. R.L. Graham, M.
Gro¨tschel and L. Lova´sz, The MITPress, Vol. II, 1996, 13311403.
[Nik] V. Nikiforov, The CycleComplete Graph Ramsey Numbers, to appear.
[NiRo1] V. Nikiforov and C.C. Rousseau, A Note on Ramsey Numbers for Books, manuscript, (2003).
[NiRo2] V. Nikiforov and C.C. Rousseau, Book Ramsey Numbers I, manuscript, (2003).
[NiRo3] V. Nikiforov and C.C. Rousseau, Large Generalized Books Are p Good, to appear in the Journal of
Combinatorial Theory, Series B, (2004).
[NiRS] V. Nikiforov, C.C. Rousseau and R.H. Schelp, Book Ramsey Numbers and QuasiRandomness,
manuscript, (2004).
[] J. Oeckermann, see [MO].
[Par1] T.D. Parsons, The Ramsey Numbers r (Pm ,Kn ), Discrete Mathematics, 6 (1973) 159162.
[Par2] T.D. Parsons, PathStar Ramsey Numbers, Journal of Combinatorial Theory, Series B, 17 (1974) 51
58.
[Par3] T.D. Parsons, Ramsey Graphs and Block Designs, I, Transactions of the American Mathematical
Society, 209 (1975) 3344.
[Par4] T.D. Parsons, Ramsey Graphs and Block Designs, Journal of Combinatorial Theory, Series A, 20
(1976) 1219.
[Par5] T.D. Parsons, Graphs from Projective Planes, Aequationes Mathematica, 14 (1976) 167189.
[Par6] T.D. Parsons, Ramsey Graph Theory, in Selected Topics in Graph Theory, (L.W. Beineke and R.J.
Wilson eds.), Academic Press, (1978) 361384.
[] T.D. Parsons, see also [FLPS].
[Piw1]* K. Piwakowski, Applying Tabu Search to Determine New Ramsey Graphs, Electronic Journal of
Combinatorics, http://www.combinatorics.org/, #R6, 3 (1996), 4 pages.
[Piw2]** K. Piwakowski, A New Upper Bound for R3(K4  e ), Congressus Numerantium, 128 (1997) 135
141.
[PR1]** K. Piwakowski and S.P. Radziszowski, 30 £ R (3,3,4) £ 31, Journal of Combinatorial Mathematics and
Combinatorial Computing, 27 (1998) 135141.
[PR2]** K. Piwakowski and S.P. Radziszowski, Towards the Exact Value of the Ramsey Number R (3,3,4),
Congressus Numerantium, 148 (2001) 161167.
[] K. Piwakowski, see also [MPR].
[] A.D. Polimeni, see [CGP, CRSPS].
[] L.M. Pretorius, see [SwPr].
[] P. Pudla´k, see [AlPu, CPR].
[Ra1]** S.P. Radziszowski, The Ramsey Numbers R (K3,K8  e ) and R (K3,K9  e ), Journal of Combinatorial
Mathematics and Combinatorial Computing, 8 (1990) 137145.
 42 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[Ra2] S.P. Radziszowski, Small Ramsey Numbers, Technical Report RITTR93009, Department of Computer
Science, Rochester Institute of Technology (1993).
[Ra3]** S.P. Radziszowski, On the Ramsey Number R (K5  e ,K5  e ), Ars Combinatoria, 36 (1993) 225232.
[RaJi] S.P. Radziszowski and Jin Xia, Paths, Cycles and Wheels in Graphs without Antitriangles, Australasian
Journal of Combinatorics, 9 (1994) 221232.
[RK1]* S.P. Radziszowski and D.L. Kreher, Search Algorithm for Ramsey Graphs by Union of Group Orbits,
Journal of Graph Theory, 12 (1988) 5972.
[RK2]** S.P. Radziszowski and D.L. Kreher, Upper Bounds for Some Ramsey Numbers R (3, k ), Journal of
Combinatorial Mathematics and Combinatorial Computing, 4 (1988) 207212.
[RK3]** S.P. Radziszowski and D.L. Kreher, On R (3, k ) Ramsey Graphs: Theoretical and Computational
Results, Journal of Combinatorial Mathematics and Combinatorial Computing, 4 (1988) 3752.
[RK4] S.P. Radziszowski and D.L. Kreher, Minimum TriangleFree Graphs, Ars Combinatoria, 31 (1991)
6592.
[RT]* S.P. Radziszowski and KungKuen Tse, A Computational Approach for the Ramsey Numbers
R (C4,Kn ), Journal of Combinatorial Mathematics and Combinatorial Computing, 42 (2002) 195207.
[RST]* S.P. Radziszowski, J. Stinehour and KungKuen Tse, Computation of the Ramsey Number
R (W5,K5 ), in preparation.
[] S.P. Radziszowski, see also [BaRT, FKR, KLR, MPR, MR1, MR2, MR3, MR4, MR5, McR, PR1,
PR2, XXER, XXR].
[Ram] F.P. Ramsey, On a Problem of Formal Logic, Proceedings of the London Mathematical Society, 30
(1930) 264286.
[Rao]* S. Rao, Applying a Genetic Algorithm to Improve the Lower Bounds of MultiColor Ramsey
Numbers, MS thesis, Department of Computer Science, Rochester Institute of Technology, 1997.
[] G. Resta, see [CPR].
[] S.W. Reyner, see [BR].
[] D.F. Reynolds, see [ExRe].
[Rob1] F.S. Roberts, Applied Combinatorics, PrenticeHall, Englewood Cliffs, 1984.
[] J.A. Roberts, see [BuRo1, BuRo2].
[] S. Roberts, see [GR].
[Rob2]* A. Robertson, New Lower Bounds for Some Multicolored Ramsey Numbers, Electronic Journal of
Combinatorics, http://www.combinatorics.org/, #R12, 6 (1999), 6 pages.
[Rob3]* A. Robertson, Difference Ramsey Numbers and Issai Numbers, Advances in Applied Mathematics, 25
(2000) 153162.
[Rob4] A. Robertson, New Lower Bounds Formulas for Multicolored Ramsey Numbers, Electronic Journal
of Combinatorics, http://www.combinatorics.org/, #R13, 9 (2002), 6 pages.
[] Y. Roditty, see [KrRod].
[Ro¨Th] V. Ro¨dl and R. Thomas, Arrangeability and Clique Subdivisions, in The Mathematics of Paul Erdo¨s
II, 236239, Algorithms and Combinatorics 14, Springer, Berlin, 1997.
[] V. Ro¨dl, see also [AlRo¨, CRST, DLR, GrRo¨, GRR1, GRR2, KoRo¨1, KoRo¨2].
[Ros] V. Rosta, On a Ramsey Type Problem of J.A. Bondy and P. Erdo¨s, I & II, Journal of Combinatorial
Theory, Series B, 15 (1973) 94120.
[] V. Rosta, see also [KR].
[] B.L. Rothschild, see [GRS].
[RoJa1] C.C. Rousseau and C.J. Jayawardene, The Ramsey Number for a Quadrilateral vs. a Complete Graph
on Six Vertices, Congressus Numerantium, 123 (1997) 97108.
 43 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[RoJa2] C.C. Rousseau and C.J. Jayawardene, Harary’s Problem for K2,k , unpublished manuscript, (1999).
[RS1] C.C. Rousseau and J. Sheehan, On Ramsey Numbers for Books, Journal of Graph Theory, 2 (1978)
7787.
[RS2] C.C. Rousseau and J. Sheehan, A Class of Ramsey Problems Involving Trees, Journal of the London
Mathematical Society (2), 18 (1978) 392396.
[] C.C. Rousseau, see also [BJYHRZ, BEFRS1, BEFRS2, BEFRS3, BEFRS4, BEFRS5, BEFRSGJ,
CLRZ, CRSPS, EFRS1, EFRS2, EFRS3, EFRS4, EFRS5, EFRS6, EFRS7, EFRS8, EFRS9, FRS1,
FRS2, FRS3, FRS4, FRS5, FRS6, FRS7, FRS8, FSR, JR1, JR2, JR3, JR4, JR5, LR1, LR2, LR3, LR4,
LRS, LRZ, NiRo1, NiRo2, NiRo3, NiRS].
[] P. Rowlinson, see [YR1, YR2, YR3].
[] A. Rucin´ski, see [GRR1, GRR2].
[San] A. Sa´nchezFlores, An Improved Bound for Ramsey Number N(3,3,3,3;2), Discrete Mathematics,
140 (1995) 281286.
[] I. Sato, see [MS].
[] R.H. Schelp, see [BaLS, BaSS, BEFRS1, BEFRS2, BEFRS3, BEFRS4, BEFRS5, BEFRSGJ, BEFS,
ChenS, EFRS1, EFRS2, EFRS3, EFRS4, EFRS5, EFRS6, EFRS7, EFRS8, EFRS9, FLPS, FRS1,
FRS2, FRS3, FRS4, FRS5, FS1, FS2, FS3, FS4, FSR, FSS1, NiRS].
[] J. Scho¨nheim, see [BS].
[SchSch1]*A. Schelten and I. Schiermeyer, Ramsey Numbers r (K3,G) for Connected Graphs G of Order Seven,
Discrete Applied Mathematics, 79 (1997) 189200.
[SchSch2] A. Schelten and I. Schiermeyer, Ramsey Numbers r (K3,G) for G @ K7  2P2 and G @ K7  3P2,
Discrete Mathematics, 191 (1998) 191196.
[] A. Schelten, see also [FSS2].
[Schi1] I. Schiermeyer, All CycleComplete Graph Ramsey Numbers r (Cm ,K6), Journal of Graph Theory, 44
(2003) 251260.
[Schi2] I. Schiermeyer, The CycleComplete Graph Ramsey Number r (C5,K7), manuscript, (2003).
[] I. Schiermeyer, see also [FSS2, SchSch1, SchSch2].
[Schu] C. U. Schulte, RamseyZahlen fu¨r Ba¨ume und Kreise, Ph.D. thesis, HeinrichHeineUniversita¨t
Du¨sseldorf, (1992).
[] S. Schuster, see[CS].
[] A. Schwenk, see [ChvS].
[Scob] M.W. Scobee, On the Ramsey Number R (m1P3,m2P3,m3P3) and Related Results, ..., MA thesis,
University of Louisville (1993).
[] R.J. Segedin, see [LorSe].
[Sha] A. Shastri, Lower Bounds for BiColored Quaternary Ramsey Numbers, Discrete Mathematics, 84
(1990) 213216.
[She1]* J.B. Shearer, Lower Bounds for Small Diagonal Ramsey Numbers, Journal of Combinatorial Theory,
Series A, 42 (1986) 302304.
[She2] J.B. Shearer, A Note on the Independence Number of Trianglefree Graphs II, Journal of Combinatorial
Theory, Series B, 53 (1991) 300307.
[] J. Sheehan, see [CRSPS, CEHMS, FRS6, FRS7, FRS8, RS1, RS2].
[] Shen YunQiu, see [LSS].
[Shi1] Shi Ling Sheng, Cube Ramsey Numbers Are Polynomial, Random Structures & Algorithms, 19
(2001) 99101.
[Shi2] Shi Ling Sheng, Upper Bounds for Ramsey Numbers, Discrete Mathematics, 270 (2003) 251265.
 44 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[ShZ1] Shi Ling Sheng and Zhang Ke Min, An Upper Bound Formula for Ramsey Numbers, manuscript,
(2001).
[ShZ2] Shi Ling Sheng and Zhang Ke Min, A Sequence of Formulas for Ramsey Numbers, manuscript,
(2001).
[ShiuLL] Shiu Wai Chee, Peter Che Bor Lam and Li Yusheng, On Some ThreeColor Ramsey Numbers,
Graphs and Combinatorics, 19 (2003) 249258.
[Sid1] A.F. Sidorenko, On Tura´n Numbers T (n , 5,4) and Number of Monochromatic 4cliques in 2colored
3graphs (in Russian), Voprosy Kibernetiki, 64 (1980) 117124.
[Sid2] A.F. Sidorenko, An Upper Bound on the Ramsey Number R (K3,G) Depending Only on the Size of
the Graph G, Journal of Graph Theory, 15 (1991) 1517.
[Sid3] A.F. Sidorenko, The Ramsey Number of an NEdge Graph Versus Triangle Is at Most 2N + 1, Journal
of Combinatorial Theory, Series B, 58 (1993) 185196.
[] M. Simonovits, see [BaSS, FSS1, FS].
[] M.J. SmugaOtto, see [AbbS].
[Sob] A. Sobczyk, Euclidian Simplices and the Ramsey Number R (4,4 ; 3), Technical Report #10, Clemson
University (1967).
[] W. Solomon, see [LorSo].
[] L. Solte´s, see [LRS].
[Song1] Song En Min, Study of Some Ramsey Numbers (in Chinese), a note (announcement of results
without proofs), Mathematica Applicata, 4(2) (1991) 6.
[Song2] Song En Min, New Lower Bound Formulas for the Ramsey Numbers N(k ,k ,...,k ;2) (in Chinese),
Mathematica Applicata, 6 (1993) suppl., 113116.
[Song3] Song En Min, An Investigation of Properties of Ramsey Numbers (in Chinese), Mathematica Applicata,
7 (1994) 216221.
[Song4] Song En Min, Properties and New Lower Bounds of the Ramsey Numbers R (p , q ;4) (in Chinese),
Journal of Huazhong University of Science and Technology, 23 (1995) suppl. II, 14.
[SYL] Song En Min, Ye Weiguo and Liu Yanwu, New Lower Bounds for Ramsey Number R (p , q ;4),
Discrete Mathematics, 145 (1995) 343346.
[] Song En Min, see also [HuSo, ZLLS].
[Spe1] J.H. Spencer, Ramsey’s Theorem  A New Lower Bound, Journal of Combinatorial Theory, Series A,
18 (1975) 108115.
[Spe2] J.H. Spencer, Asymptotic Lower Bounds for Ramsey Functions, Discrete Mathematics, 20
(1977/1978) 6976.
[] J.H. Spencer, see also [BES, GRS].
[Spe3]* T. Spencer, University of Nebraska at Omaha, personal communication (1993), and, Upper Bounds
for Ramsey Numbers via Linear Programming, preprint, (1994).
[Stahl] S. Stahl, On the Ramsey Number R (F ,Km ) where F is a Forest, Canadian Journal of Mathematics,
27 (1975) 585589.
[] R.G. Stanton, see [KaSt].
[Stat] W. Staton, Some Ramseytype Numbers and the Independence Ratio, Transactions of the American
Mathematical Society, 256 (1979) 353370.
[] A. Steger, see [McS].
[] J. Stinehour, see [RST].
[Stev] S. Stevens, Ramsey Numbers for Stars Versus Complete Multipartite Graphs, Congressus Numerantium,
73 (1990) 6371.
[] M.J. Stewart, see [CRSPS].
 45 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[Stone] J.C. Stone, Utilizing a Cancellation Algorithm to Improve the Bounds of R (5, 5), (1996),
http://oas.okstate.edu/ojas/jstone.htm. This paper claims incorrectly that R (5,5) = 50.
[SL]* Su Wenlong and Luo Haipeng, Prime Order Cyclic Graphs and New Lower Bounds for Three Classical
Ramsey Numbers R (4, n ) (in Chinese), Journal of Mathematical Study, 31, 4 (1998) 442446.
[SLLL]* Su Wenlong, Luo Haipeng, Li Guiqing and Li Qiao, Lower Bounds of Ramsey Numbers Based on
Cubic Residues, Discrete Mathematics, 250 (2002) 197209.
[SLL1]* Su Wenlong, Luo Haipeng and Li Guiqing, Two Lower Bounds of Classical 2color Ramsey
Numbers R (3, q ) (in Chinese), Journal of Guangxi University for Nationalities, 5, 1 (1999) 14.
[SLL2]* Su Wenlong, Luo Haipeng and Li Qiao, New Lower Bounds of Classical Ramsey Numbers R (4,12),
R (5,11) and R (5,12), Chinese Science Bulletin, 43, 6 (1998) 528.
[SLL3]* Su Wenlong, Luo Haipeng and Li Qiao, New Lower Bounds for Seven Classical Ramsey Numbers
R (k , l ) (in Chinese), Journal of Systems Science and Mathematical Sciences, 20, 1 (2000) 5557.
[SLZL]* Su Wenlong, Luo Haipeng, Zhang Zhengyou and Li Guiqing, New Lower Bounds of Fifteen Classical
Ramsey Numbers, Australasian Journal of Combinatorics, 19 (1999) 9199.
[] Su Wenlong, see also [LSL, LSS, LSZL, ZSL].
[Sud1] B. Sudakov, A Note on Odd CycleComplete Graph Ramsey Numbers, Electronic Journal of Combinatorics,
http://www.combinatorics.org/, #N1, 9 (2002), 4 pages.
[Sud2] B. Sudakov, Large Kr free Subgraphs in Ks free Graphs and Some Other RamseyType Problems,
preprint.
[] B. Sudakov, see also [AlKS, KoSu].
[] A. Sudan, see [GGS].
[SuBa1] Surahmat and E.T. Baskoro, On the Ramsey Number of a Path or a Star versus W4 or W5, Proceedings
of the 12th Australasian Workshop on Combinatorial Algorithms, Bandung, Indonesia, July 14
17 (2001) 174179.
[SuBa2] Surahmat and E.T. Baskoro, The Ramsey Number of Linear Forest versus Wheel, paper presented at
the 13th Australasian Workshop on Combinatorial Algorithms, Fraser Island, Queensland, Australia,
July 710, 2002.
[SuBB1] Surahmat, E.T. Baskoro and H.J. Broersma, The Ramsey Numbers of Large Starlike Trees versus
Large Odd Wheels, Technical Report #1621, Faculty of Mathematical Sciences, University of
Twente, The Netherlands, (2002).
[SuBB2] Surahmat, E.T. Baskoro and H.J. Broersma, The Ramsey Numbers of Large Cycles versus Small
Wheels, Technical Report #1634, Faculty of Mathematical Sciences, University of Twente, The Netherlands,
(2002).
[] Surahmat, see also [BaSu, BSNM].
[SwPr] C.J. Swanepoel and L.M. Pretorius, Upper Bounds for a Ramsey Theorem for Trees, Graphs and
Combinatorics, 10 (1994) 377382.
[] M.M. Sweet, see [FreSw].
[] E. Szemere´di, see [AKS, CRST].
[] R. Thomas, see [Ro¨Th].
[Tho] A. Thomason, An Upper Bound for Some Ramsey Numbers, Journal of Graph Theory, 12 (1988)
509517.
[] P.W. Tingley, see [HaŁT].
[] C.A. Tovey, see [CET].
[Tr] Trivial results.
[] W.T. Trotter Jr., see [CRST].
[Tse1]* KungKuen Tse, On the Ramsey Number of the Quadrilateral versus the Book and the Wheel, Australasian
Journal of Combinatorics, 27 (2003) 163167.
 46 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[Tse2]* KungKuen Tse, Kean University, personal communication, (2004).
[] KungKuen Tse, see also [BaRT, RST, RT].
[] Z. Tuza, see [GT].
[] L. Volkmann, see [GV].
[Walk] K. Walker, Dichromatic Graphs and Ramsey Numbers, Journal of Combinatorial Theory, 5 (1968)
238243.
[Wall] W.D. Wallis, On a Ramsey Number for Paths, Journal of Combinatorics, Information & System Sciences,
6 (1981) 295296.
[Wan] Wan Honghui, Upper Bounds for Ramsey Numbers R (3, 3, . . . , 3) and Schur Numbers, Journal of
Graph Theory, 26 (1997) 119122.
[] Wang Gongben, see [WW, WWY1, WWY2].
[WW]* Wang Qingxian and Wang Gongben, New Lower Bounds of Ramsey Numbers r (3, q ) (in Chinese),
Acta Scientiarum Naturalium, Universitatis Pekinensis, 25 (1989) 117121.
[WWY1]* Wang Qingxian, Wang Gongben and Yan Shuda, A Search Algorithm And New Lower Bounds for
Ramsey Numbers r (3, q ), preprint, (1994).
[WWY2]* Wang Qingxian, Wang Gongben and Yan Shuda, The Ramsey Numbers R (K3,Kq  e ) (in Chinese),
Beijing Daxue Xuebao Ziran Kexue Ban, 34 (1998) 1520.
[] Wang Zhi Jian, see [LiWa1, LiWa2].
[Wh] E.G. Whitehead, The Ramsey Number N(3,3,3,3 ; 2), Discrete Mathematics, 4 (1973) 389396.
[] E.R. Williams, see [AbbW].
[] R.M. Wilson, see [FraWi].
[] A. Woldar, see [LaWo1, LaWo2].
[XZ]* Xie Jiguo and Zhang Xiaoxian, A New Lower Bound for Ramsey Number r (3,13) (in Chinese),
Journal of Lanzhou Railway Institute, 12 (1993) 8789.
[] Xie Zheng, see [XX1, XX2, XXER, XXR].
[Xu] Xu Xiaodong, personal communication, (2004).
[XX1]* Xu Xiaodong and Xie Zheng, A Constructive Approach for the Lower Bounds on the Ramsey
Numbers r (k , l ), manuscript, (2002).
[XX2] Xu Xiaodong and Xie Zheng, A Constructive Approach for the Lower Bounds on Multicolor Ramsey
Numbers, manuscript, (2002).
[XXER]* Xu Xiaodong, Xie Zheng, G. Exoo and S.P. Radziszowski, Constructive Lower Bounds on Classical
Multicolor Ramsey Numbers, Electronic Journal of Combinatorics, http://www.combinatorics.org/,
#R35, 11 (2004), 24 pages.
[XXR] Xu Xiaodong, Xie Zheng and S.P. Radziszowski, A Constructive Approach for the Lower Bounds on
the Ramsey Numbers R (s , t ), to appear in the Journal of Graph Theory, (2004).
[] J. Yackel, see [GY].
[] Yan Shuda, see [WWY1, WWY2].
[YHZ1] Yang Jian Sheng, Huang Yi Ru and Zhang Ke Min, The Value of the Ramsey Number R (Cn ,K4 ) is
3(n  1) + 1 (n ³ 4), Australasian Journal of Combinatorics, 20 (1999) 205206.
[YHZ2] Yang Jian Sheng, Huang Yi Ru and Zhang Ke Min, R (C6,K5 ) = 21 and R (C7,K5 ) = 25, European
Journal of Combinatorics, 22 (2001) 561567.
[] Yang Jian Sheng, see also [BJYHRZ].
[YY]** Yang Yuansheng, On the Third Ramsey Numbers of Graphs with Six Edges, Journal of Combinatorial
Mathematics and Combinatorial Computing, 17 (1995) 199208.
[YH]* Yang Yuansheng and G.R.T. Hendry, The Ramsey Number r (K1 + C4,K5  e ), Journal of Graph
Theory, 19 (1995) 1315.
 47 
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
[YR1]** Yang Yuansheng and P. Rowlinson, On the Third Ramsey Numbers of Graphs with Five Edges,
Journal of Combinatorial Mathematics and Combinatorial Computing, 11 (1992) 213222.
[YR2]* Yang Yuansheng and P. Rowlinson, On Graphs without 6Cycles and Related Ramsey Numbers,
Utilitas Mathematica, 44 (1993) 192196.
[YR3]* Yang Yuansheng and P. Rowlinson, The Third Ramsey Numbers for Graphs with at Most Four
Edges, Discrete Mathematics, 125 (1994) 399406.
[] Ye Weiguo, see [SYL].
[Yu1]* Yu Song Nian, A Computer Assisted Number Theoretical Construction of (3, k )Ramsey Graphs,
Annales Universitatis Scientiarum Budapestinensis, Sect. Comput., 10 (1989) 3544.
[Yu2]* Yu Song Nian, Maximal Trianglefree Circulant Graphs and the Function K(c ) (in Chinese), Journal
of Shanghai University, Natural Science, 2 (1996) 678682.
[Zali] Zang Wenan and Li Yusheng, The Independence Number of Graphs with a Forbidden Cycle and
Ramsey Numbers, Journal of Combinatorial Optimization, 7 (2003) 353359.
[] Zang Wenan, see [LRZ, LZ].
[Zeng] Zeng Wei Bin, Ramsey Numbers for Triangles and Graphs of Order Four with No Isolated Vertex,
Journal of Mathematical Research & Exposition, 6 (1986) 2732.
[ZZ] Zhang Ke Min and Zhang Shu Sheng, Some TreeStars Ramsey Numbers, Proceedings of the Second
Asian Mathematical Conference 1995, 287291, World Sci. Publishing, River Edge, NJ, 1998.
[] Zhang Ke Min, see also [BJYHRZ, ChenZZ1, ChenZZ2, ChenZZ3, HZ1, HZ2, MZ, ShZ1, ShZ2,
YHZ1, YHZ2].
[] Zhang Shu Sheng, see [ZZ].
[] Zhang Xiaoxian, see [XZ].
[] Zhang Yuming, see [CLRZ].
[] Zhang Yunqing, see [ChenZZ1, ChenZZ2, ChenZZ3].
[ZSL] Zhang Zhengyou, Su Wenlong and Luo Haipeng, Edge Coloration of Complete Graph K97 and New
Lower Bounds of Four Ramsey Numbers (in Chinese), Application Research of Computers, 18, 6
(2001) 2931, 57.
[] Zhang Zhengyou, see also [LSZL, SLZL].
[ZLLS] Zhang Zhongfu, Liu Linzhong, Li Jinwen and Song En Min, Some Properties of Ramsey Numbers,
Applied Mathematics Letters, 16 (2003) 11871193.
[Zhou1] Zhou Huai Lu, Some Ramsey Numbers for Graphs with Cycles (in Chinese), Mathematica Applicata,
6 (1993) 218.
[Zhou2] Zhou Huai Lu, The Ramsey Number of an Odd Cycle with Respect to a Wheel (in Chinese), Journal
of Mathematics, Shuxue Zazhi (Wuhan), 15 (1995) 119120.
[Zhou3] Zhou Huai Lu, On BookWheel Ramsey Number, Discrete Mathematics, 224 (2000) 239249.
Out of 403 references gathered above, 327 appeared in 58 different periodicals, among
which most articles were published in: Discrete Mathematics 49, Journal of Combinatorial
Theory (old, Series A and B) 42, Journal of Graph Theory 41, and Ars Combinatoria 19. The
results of 93 references depend on computer algorithms.
 48  
lisp in small pieces pdf
20151106 21:00:31The book is in two parts. The first starts from a simple evaluation function and enriches it with multiple name spaces, continuations and sideeffects with commented variants, while at the same time ... 
DPM(Defomable Parts Model) 源码分析
20140123 09:58:00DPM(Deformable Parts Model)原理(一) 原文：http://blog.csdn.net/ttransposition/article/details/12966521 DPM(Deformable Parts Model) Reference: Object detection with discriminatively ... 
Lua: Good, bad, and ugly parts
20150415 10:46:39Lua: Good, bad, and ugly parts from :http://notebook.kulchenko.com/programming/luagooddifferentbadanduglyparts 25 Mar 2012 I have been programming in Lua for about 9 months and it 
16 CFR Part 1510 小零件（Small part）识别方法  完整英文版（4页）
20210319 17:05:4716 CFR Part 1510  METHOD FOR IDENTIFYING TOYS AND OTHER ARTICLES INTENDEDFOR USE BY CHILDREN UNDER 3 YEARS OF AGE WHICH PRESENT CHOKING,ASPIRATION, OR INGESTION HAZARDS BECAUSE OF SMALL PARTS(识别3岁... 
Unit 2: Reading The Parts of Speech
20141004 17:02:01In this unit you will learn the eight basic parts of speech and how they function in English. To become a better writer and editor of your writing, you should know the fundamentals of English prose an 
Removed the parts where the document is stating that EQ should be written against parent elements.
20201209 12:26:35<div><p>See #11 (especially the lower parts). I generated the HTML file with the latest bikeshed version, which changed the ... 
【Leetcode】725. Split Linked List in Parts
20180222 21:44:00Given a (singly) linked list with head node root, write a function to split the linked list into k consecutive linked list "parts". The length of each part should be as equal as possible: no two part... 
python中skimage包的小优化(1)：模仿remove_small_objects（)函数去除图片边缘不感兴趣区域
20180510 22:34:11python模仿skimage包中的remove_small_objects（)函数实现去除边缘噪点 图片难免会有噪声，python的skimage包提供了名为morphology的子模块，可以通过调用该模块的remove_small_objects（)进行图片去噪。具体使用... 
Small：轻巧的跨平台插件化框架
20160723 23:32:20Small 世界那么大，组件那么小。Small，做最轻巧的跨平台插件化框架。 支持平台：Android API 15(4.0.3)+iOS 7.0+ 功能 完美内置 所有插件支持内置于宿主包中 高度透明 插件编码、布局... 
使用CSS Shadow Parts在Shadow DOM中进行样式化
20200628 11:44:51Safari 13.1刚刚提供了对CSS Shadow Parts的支持。 这意味着Chrome，Edge，Opera，Safari和Firefox现在支持::part()选择器。 我们将看到为什么它有用，但首先回顾一下影子DOM封装… 影子DOM封装的好处 我在... 
DPM(Deformable Parts Model)原理及代码分析
20160111 20:37:48DPM(Deformable Parts Model) Reference: Object detection with discriminatively trained partbased models. IEEE Trans. PAMI, 32(9):1627–1645, 2010. "Support Vector Machines for MultipleInstance 
DPM(Defomable Parts Model) 源码分析训练（三）
20151124 09:42:26Parts Model) 源码分析训练（三） DPM(Defomable Parts Model)原理 首先调用格式： example: pascal('person', 2); % train and evaluate a 2 component person model pascal_train.m 
DPM(Defomable Parts Model) 源码分析检测（二）
20131022 21:17:57DPM(Defomable Parts Model)原理 首先声明此版本为V3.1。因为和论文最相符。V4增加了模型数由2个增加为6个，V5提取了语义特征。源码太长纯代码应该在2K+,只选取了核心部分代码 demo.m function demo() ... 
Make Projects: Small Form Factor PCs
20081226 20:22:00版权声明：原创作品，允许转载，转载时请务必以超链接形式...http://blog.csdn.net/topmvp  topmvpShoebox sized and smaller, small form factor PCs can pack as much computing muscle as anything from a PDA to 
Building Maintainable Softwarejava篇之Keep Your Codebase Small
20160216 22:36:56Building Maintainable Softwarejava篇之Keep Your Codebase Small Program complexity grows until it exceeds the capability of the programmer who must maintain it. —7th Law of Computer ... 
Building Maintainable Softwarejava篇之Keep Unit Interfaces Small
20160213 13:18:41Building Maintainable Softwarejava篇之Keep Unit Interfaces Small Bunches of data that hang around together really ought to be made into their own object. —Marti 
skimage.morphology.remove_small_objects移除面积小的连通区域无效
20200628 15:26:39使用remove_small_objects函数移除二值图像中小的连通域时发现用后并没有得到所期待的结果，经检索发现类似的问题： ... 
Guidelines for the Design of Small Sewage Treatment Plants
20151023 22:37:44be of the type that allows the underwater parts to be maintained without the need of shutting down the tank. 4.25 Adjustable weirs should be used for sedimentation tanks... 
Using Reporting Services SharePoint Web Parts in SQL Server 2000 Reporting Services Service Pack 2
20051103 17:02:00Using Reporting Services SharePoint Web Parts in SQL Server 2000 Reporting Services Service Pack 2Published: April 25, 2005SQL Server Technical ArticleAuthor: Duncan Smith, IdentityMine Technical Revi 
Black  Varsity Red  White The new 23 strong cold air temperature in most parts of incoming 61
20100426 04:55:00train collector html simple template model The new 23 strong cold air temperature in most parts of incoming 610 ℃ _ News _ Tencent Network The new 23 strong cold air tem