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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。
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• <div><p><img alt="inbox1" src="https://img-blog.csdnimg.cn/img_convert/8143c8f8375aee6b8dc83b1abdc7d3dc.png" /></p>该提问来源于开源项目：boukestam/inbox-in-gmail</p></div>
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 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。
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• 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 ...
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• SMALL_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 NSL-32SR3 VACTROL_5PIN适用于VTL5C3 / 2和类似的五针vactrol Radio Music的YAMAICHI PJS008U-3000-
• ## Small Ramsey Numbers

千次阅读 2006-09-13 10:02:00
Small Ramsey NumbersStanisław P. RadziszowskiDepartment of Computer ScienceRochester Institute of TechnologyRochester, NY 14623, spr@cs.rit.eduSubmitted: June 11, 1994; Accepted: July 3, 1994Revision
Small Ramsey Numbers
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, RIT-TR-93-009 [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
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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
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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 m-color 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.
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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.
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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
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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
triangle-free 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.
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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 triangle-free 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) Two-color 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 triangle-free graphs with independence k on W(k 3/ 2 ) vertices [Alon2, CPR].
(o) Other general and asymptotic results on triangle-free graphs in relation to the case of
R (3, k ) [AKS, Alon2, CCD, CPR, Gri, FrLo, Loc, She2].
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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
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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 .
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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) ).
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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]
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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 triangle-free 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]
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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¨].
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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 isolate-free 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 isolate-free 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]
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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 star-like 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]
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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]
Star-like 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
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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)(m-1) + 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
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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 isolate-free 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 isolate-free graphs G on 4 vertices.
[BES] Study of Ramsey numbers for multiple copies of graphs.
[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 NP-hard.
[-] Special cases of multicolor results listed in section 5.
[-] See also surveys listed in section 7.
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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].
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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´nchez-Flores [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).
Off-Diagonal 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 3-color
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 not-too-distance 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]
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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]
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(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].
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- 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).
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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
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[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 3-uniform 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 isolate-free 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.
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[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 isolate-free graphs with at most 4 vertices.
Some of the missing cases completed in [KM2].
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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 zero-sum 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 (1903-1930) as a person.
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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, zero-sum 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.
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Mathematics, 40 (1982) 11-20.
[Bu4] S.A. Burr, Diagonal Ramsey Numbers for Small Graphs, Journal of Graph Theory, 7 (1983) 57-69.
[Bu5] S.A. Burr, Ramsey Numbers Involving Powers, Ars Combinatoria, 15 (1983) 163-168.
[Bu6] S.A. Burr, Determining Generalized Ramsey Numbers is NP-Hard, Ars Combinatoria, 17 (1984) 21-
25.
[Bu7] S.A. Burr, What Can We Hope to Accomplish in Generalized Ramsey Theory?, Discrete Mathematics,
67 (1987) 215-225.
[Bu8] S.A. Burr, On the Ramsey Numbers r (G, nH ) and r (nG , nH ) When n Is Large, Discrete Mathematics,
65 (1987) 215-229.
[Bu9] S.A. Burr, On Ramsey Numbers for Large Disjoint Unions of Graphs, Discrete Mathematics, 70
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[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 Ramsey-Theoretic Result of Chva´tal, Journal of Graph
Theory, 7 (1983) 39-51.
[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) 193-203.
[BEFRS2] S.A. Burr, P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Ramsey Numbers for the Pair
Sparse Graph-Path 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) 395-409.
[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 Graph-Graph of Limited Degree, in Graph Theory with Applications to Algorithms
and Computer Science, (Y. Alavi et al. eds.), John Wiley & Sons, New York, (1985) 163-174.
[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) 79-89.
[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) 1-6.
[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) 115-118.
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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
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[BF] S.A. Burr and R.J. Faudree, On Graphs G for Which All Large Trees Are G-good, Graphs and Combinatorics,
9 (1993) 305-313.
[BG] S.A. Burr and J.W. Grossman, Ramsey Numbers of Graphs with Long Tails, Discrete Mathematics,
41 (1982) 223-227.
[BuRo1] S.A. Burr and J.A. Roberts, On Ramsey Numbers for Stars, Utilitas Mathematica, 4 (1973) 217-220.
[BuRo2] S.A. Burr and J.A. Roberts, On Ramsey Numbers for Linear Forests, Discrete Mathematics, 8 (1974)
245-250.
[Bush] L.E. Bush, The William Lowell Putnam Mathematical Competition, American Mathematical Monthly,
60 (1953) 539-542.
[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) 381-387.
[Caro] Y. Caro, Zero-Sum Problems - A Survey, Discrete Mathematics, 152 (1996) 93-113.
[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) 51-56.
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Graphs, Journal of Graph Theory, 4 (1980) 233-239.
[CRSPS] G. Chartrand, C.C. Rousseau, M.J. Stewart, A.D. Polimeni and J. Sheehan, On Star-Book Ramsey
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Graphs, (Kalamazoo, MI 1980), John Wiley & Sons, (1981) 203-214.
[CS] G. Chartrand and S. Schuster, On the existence of specified cycles in complementary graphs, Bulletin
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Theory, Series B, 57 (1993) 138-149.
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University, Natural Science, 25 (2002) 244-246.
[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) 281-285.
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(R. Bari and F. Harary eds.), Springer LNM 406, Berlin, (1974) 236-241.
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[ChGra2] F.R.K. Chung and R.L. Graham, Erdo¨s on Graphs, His Legacy of Unsolved Problems, A K Peters,
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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
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[Chv] V. Chva´tal, Tree-Complete 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,
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[CH2] V. Chva´tal and F. Harary, Generalized Ramsey Theory for Graphs, III. Small Off-Diagonal
Numbers, Pacific Journal of Mathematics, 41 (1972) 335-345.
[CH3] V. Chva´tal and F. Harary, Generalized Ramsey Theory for Graphs, I. Diagonal Numbers, Periodica
Mathematica Hungarica, 3 (1973) 115-124.
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Bounded Maximum Degree, Journal of Combinatorial Theory, Series B, 34 (1983) 239-243.
[ChvS] V. Chva´tal and A. Schwenk, On the Ramsey Number of the Five-Spoked Wheel, in Graphs and
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Journal of Graph Theory, 13 (1989) 7-15.
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[-] R. Cleve, see [CCD].
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Mathematical Society, Series A, 19 (1975) 252-256.
[CL2] E.J. Cockayne and P.J. Lorimer, On Ramsey Graph Numbers for Stars and Stripes, Canadian
Mathematical Bulletin, 18 (1975) 31-34.
[CPR] B. Codenotti, P. Pudla´k and G. Resta, Some Structural Properties of Low-Rank Matrices Related to
Computational Complexity, Theoretical Computer Science, 235 (2000) 89-107.
[CoPC] Special issue on Ramsey theory of Combinatorics, Probability and Computing, 12 (2003), Numbers 5
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[-] P. Dagum, see [CCD].
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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) 63-75.
[Eng] A. Engstro¨m, A Note on Two Multicolor Ramsey Numbers, manuscript, (2003).
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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,
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[Erd2] P. Erdo¨s, On the Combinatorial Problems Which I Would Most Like to See Solved, Combinatorica,
1 (1981) 25-42.
[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) 250-264.
[EFRS2] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, On Cycle-Complete Graph Ramsey
Numbers, Journal of Graph Theory, 2 (1978) 53-64.
[EFRS3] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Ramsey Numbers for Brooms, Congressus
Numerantium, 35 (1982) 283-293.
[EFRS4] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Multipartite Graph-Sparse Graph Ramsey
Numbers, Combinatorica, 5 (1985) 311-318.
[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) 227-233.
[EFRS6] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Extremal Theory and Bipartite Graph-Tree
Ramsey Numbers, Discrete Mathematics, 72 (1988) 103-112.
[EFRS7] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, The Book-Tree Ramsey Numbers, Scientia,
Series A: Mathematical Sciences, Valparaı´so, Chile, 1 (1988) 111-117.
[EFRS8] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Multipartite Graph-Tree Graph Ramsey
Numbers, in Graph Theory and Its Applications: East and West, Proceedings of the First China-USA
International Graph Theory Conference, Annals of the New York Academy of Sciences, 576 (1989)
146-154.
[EFRS9] P. Erdo¨s, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Ramsey Size Linear Graphs, Combinatorics,
Probability and Computing, 2 (1993) 389-399.
[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) 515-527.
[-] 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) 5-11.
[Ex2]* G. Exoo, Constructing Ramsey Graphs with a Computer, Congressus Numerantium, 59 (1987) 31-36.
[Ex3]* G. Exoo, Applying Optimization Algorithm to Ramsey Problems, in Graph Theory, Combinatorics,
Algorithms, and Applications (Y. Alavi ed.), SIAM Philadelphia, (1989) 175-179.
[Ex4]* G. Exoo, A Lower Bound for R (5, 5), Journal of Graph Theory, 13 (1989) 97-98.
[Ex5]* G. Exoo, On Two Classical Ramsey Numbers of the Form R (3, n ), SIAM Journal of Discrete
Mathematics, 2 (1989) 488-490.
[Ex6]* G. Exoo, A Lower Bound for r (K5 - e ,K5), Utilitas Mathematica, 38 (1990) 187-188.
[Ex7]* G. Exoo, On the Three Color Ramsey Number of K4 - e , Discrete Mathematics, 89 (1991) 301-305.
[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) 109-114. Constructions available at
http://ginger.indstate.edu/ge/RAMSEY.
[Ex16]* G. Exoo, Indiana State University, personal communication (2002-2004). 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) 277-286.
[EHM2] G. Exoo, H. Harborth and I. Mengersen, On Ramsey Number of K2,n , in Graph Theory, Combinatorics,
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[-] G. Exoo, see also [CEHMS, XXER].
[FLPS] R.J. Faudree, S.L. Lawrence, T.D. Parsons and R.H. Schelp, Path-Cycle Ramsey Numbers, Discrete
Mathematics, 10 (1974) 269-277.
[FM]** R.J. Faudree and B.D. McKay, A Conjecture of Erdo¨s and the Ramsey Number r (W6), Journal of
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[FRS1] R.J. Faudree, C.C. Rousseau and R.H. Schelp, All Triangle-Graph Ramsey Numbers for Connected
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[FRS2] R.J. Faudree, C.C. Rousseau and R.H. Schelp, Studies Related to the Ramsey Number r (K5 - e ), in
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[FRS3] R.J. Faudree, C.C. Rousseau and R.H. Schelp, Generalizations of the Tree-Complete Graph Ramsey
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York, (1985) 117-126.
[FRS4] R.J. Faudree, C.C. Rousseau and R.H. Schelp, Small Order Graph-Tree Ramsey Numbers, Discrete
Mathematics, 72 (1988) 119-127.
[FRS5] R.J. Faudree, C.C. Rousseau and R.H. Schelp, A Good Idea in Ramsey Theory, in Graph Theory,
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[FRS6] R.J. Faudree, C.C. Rousseau and J. Sheehan, More from the Good Book, in Proceedings of the Ninth
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[FRS7] R.J. Faudree, C.C. Rousseau and J. Sheehan, Strongly Regular Graphs and Finite Ramsey Theory,
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[FRS8] R.J. Faudree, C.C. Rousseau and J. Sheehan, Cycle-Book Ramsey Numbers, Ars Combinatoria, 31
(1991) 239-248.
[FS1] R.J. Faudree and R.H. Schelp, All Ramsey Numbers for Cycles in Graphs, Discrete Mathematics, 8
(1974) 313-329.
[FS2] R.J. Faudree and R.H. Schelp, Path Ramsey Numbers in Multicolorings, Journal of Combinatorial
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[FS3] R.J. Faudree and R.H. Schelp, Ramsey Numbers for All Linear Forests, Discrete Mathematics, 16
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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
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[FSR] R.J. Faudree, R.H. Schelp and C.C. Rousseau, Generalizations of a Ramsey Result of Chva´tal, in
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(Kalamazoo, MI 1980), John Wiley & Sons, (1981) 351-361.
[FSS1] R.J. Faudree, R.H. Schelp and M. Simonovits, On Some Ramsey Type Problems Connected with
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[FSS2] R.J. Faudree, A. Schelten and I. Schiermeyer, The Ramsey Number r (C7,C7,C7), Discussiones
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Problems, Journal of Graph Theory, 16 (1992) 25-50.
[-] 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
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[Fo] J. Folkman, Notes on the Ramsey Number N(3,3,3,3), Journal of Combinatorial Theory, Series A, 16
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[FraWi] P. Frankl and R.M. Wilson, Intersection Theorems with Geometric Consequences, Combinatorica, 1
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[Fra1] K. Fraughnaugh Jones, Independence in Graphs with Maximum Degree Four, Journal of Combinatorial
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[Fra2] K. Fraughnaugh Jones, Size and Independence in Triangle-Free Graphs with Maximum Degree
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[Fre] H. Fredricksen, Schur Numbers and the Ramsey Numbers N(3,3,...,3 ; 2), Journal of Combinatorial
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[FreSw]* H. Fredricksen and M.M. Sweet, Symmetric Sum-Free Partitions and Lower Bounds for Schur
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[Gi2] G. Giraud, Minoration de certains nombres de Ramsey binaires par les nombres de Schur ge´ne´ralise´s,
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[Gi3] G. Giraud, Nouvelles majorations des nombres de Ramsey binaires-bicolores, C.R. Acad. Sc. Paris,
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[Gi4] G. Giraud, Majoration du nombre de Ramsey ternaire-bicolore en (4,4), C.R. Acad. Sc. Paris, Se´ries
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[Gi5] G. Giraud, Une minoration du nombre de quadrangles unicolores et son application a la majoration
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[-] R.J. Gould, see also [BEFRSGJ, CGP].
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[-] R.L. Graham, see also [ChGra1, ChGra2, EG].
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[GG] R.E. Greenwood and A.M. Gleason, Combinatorial Relations and Chromatic Graphs, Canadian Journal
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[GH] U. Grenda and H. Harborth, The Ramsey Number r (K3,K7 - e ), Journal of Combinatorics, Information
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[Gri] J.R. Griggs, An Upper Bound on the Ramsey Numbers R (3,k ), Journal of Combinatorial Theory,
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[GR]** C. Grinstead and S. Roberts, On the Ramsey Numbers R (3,8) and R (3,9), Journal of Combinatorial
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THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
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THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
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- 47 -
THE ELECTRONIC JOURNAL OF COMBINATORICS (2004), DS1.10
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YHZ1, YHZ2].
[-] Zhang Shu Sheng, see [ZZ].
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[-] Zhang Yunqing, see [ChenZZ1, ChenZZ2, ChenZZ3].
[ZSL] Zhang Zhengyou, Su Wenlong and Luo Haipeng, Edge Coloration of Complete Graph K97 and New
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(2001) 29-31, 57.
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[ZLLS] Zhang Zhongfu, Liu Linzhong, Li Jinwen and Song En Min, Some Properties of Ramsey Numbers,
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[Zhou1] Zhou Huai Lu, Some Ramsey Numbers for Graphs with Cycles (in Chinese), Mathematica Applicata,
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[Zhou2] Zhou Huai Lu, The Ramsey Number of an Odd Cycle with Respect to a Wheel (in Chinese), Journal
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[Zhou3] Zhou Huai Lu, On Book-Wheel Ramsey Number, Discrete Mathematics, 224 (2000) 239-249.
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 - 
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