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15
All Ramsey numbers r(K 3 ,G) for connected graphs of order 9
, 1998
"... We determine the Ramsey numbers r(K 3 , G) for all 261080 connected graphs of order 9 and further Ramsey numbers of this type for some graphs of order up to 12. Almost all of them were determined by computer programs which are based on a program for generating maximal trianglefree graphs. ..."
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We determine the Ramsey numbers r(K 3 , G) for all 261080 connected graphs of order 9 and further Ramsey numbers of this type for some graphs of order up to 12. Almost all of them were determined by computer programs which are based on a program for generating maximal trianglefree graphs.
A Conjecture of Erdös the Ramsey Number r(W6)
 J. Combinatorial Math. and Combinatorial Computing
, 1996
"... It was conjectured by Paul Erdos that if G is a graph with chromatic number at least k; then the diagonal Ramsey number r(G) r(K k ). That is, the complete graph K k has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by ve ..."
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It was conjectured by Paul Erdos that if G is a graph with chromatic number at least k; then the diagonal Ramsey number r(G) r(K k ). That is, the complete graph K k has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by verifying that r(W 6 ) = 17; where W 6 is the wheel with 6 vertices, since it is well known that r(K 4 ) = 18. Computational techniques are used to determine r(W 6 ) as well as the Ramsey numbers for other pairs of small order wheels. 1 Introduction The following well known conjecture is due to Paul Erdos. CONJECTURE 1 If G is a graph with chromatic number Ø(G) k; then the Ramsey number r(G) r(K k ): The strong form of the Erdos conjecture is that if Ø(G) k; and G does not contain a copy of K k ; then r(G) ? r(K k ). For k = 3 it is trivial to verify this stronger conjecture. If G 6' K 3 and Ø(G) 3; then G has at least 4 vertices. Thus r(G) ? 6 = r(K 3 ); since neither the graph K 3 [K...
New Bounds on Some Ramsey Numbers
"... We derive a new upper bound of 26 for the Ramsey number R(K5 − P3, K5), lowering the previous upper bound of 28. This leaves 25 ≤ R(K5 − P3, K5) ≤ 26, improving on one of the three remaining open cases in Hendry’s table, which listed Ramsey numbers for pairs of graphs (G, H) with G and H having f ..."
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We derive a new upper bound of 26 for the Ramsey number R(K5 − P3, K5), lowering the previous upper bound of 28. This leaves 25 ≤ R(K5 − P3, K5) ≤ 26, improving on one of the three remaining open cases in Hendry’s table, which listed Ramsey numbers for pairs of graphs (G, H) with G and H having five vertices. We also show, with the help of a computer, that R(B2, B6) = 17 and R(B2, B7) = 18 by full enumeration of (B2, B6)good graphs and (B2, B7)good graphs, where Bn is the book graph with n triangular pages.
Ramsey Numbers of K m versus (n,k)graphs and the Local Density of Graphs not Containing a K m
"... In this paper generalized Ramsey numbers of complete graphs Km versus the set h; n; ki of (n; k)graphs are investigated. The value of r(Km ; hn; ki) is given in general for (relative to n) values of k small compared to n using a correlation with Tur'an numbers. These generalized Ramsey numbers can ..."
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In this paper generalized Ramsey numbers of complete graphs Km versus the set h; n; ki of (n; k)graphs are investigated. The value of r(Km ; hn; ki) is given in general for (relative to n) values of k small compared to n using a correlation with Tur'an numbers. These generalized Ramsey numbers can be used to determine the local densities of graphs not containing a subgraph Km . 1 Introduction Let m, l, n and k be positive integers with 0 l i m 2 j and 0 k i n 2 j and let hn; ki denote the set of all (n; k)graphs, i.e. the set of all graphs with n vertices and k edges. The Ramsey number r(hm; li; hn; ki) is defined as the smallest integer p such that in every redgreen coloring of the edges of the complete graph K p a green (m; l)graph or a red (n; k)graph occurs, i.e. a green graph with m vertices and l edges or a red graph with n vertices and k edges. Note that r(hm; li; hn; ki) is the classical Ramsey number r(Km ; K n ) if l = i m 2 j and k = i n 2 j ...
Recent Results in Computational Ramsey Theory
, 1992
"... Since its conception in 1930, classical and generalized Ramsey theory has been widely studied. Despite this, very little progress has been made, due to the intractability of the problems encountered. In the past year, several new exact values and improved bounds have been discovered for Ramsey nu ..."
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Since its conception in 1930, classical and generalized Ramsey theory has been widely studied. Despite this, very little progress has been made, due to the intractability of the problems encountered. In the past year, several new exact values and improved bounds have been discovered for Ramsey numbers, by using a hybrid approach of clever mathematics and large scale computer search. The validity of this increasingly important method of analysis raises some important questions which must be resolved by mathematicians and computer scientists. 1 1 Introduction 1.1 Ramsey's Theorem The Ramsey number R(k; l) can be dened as the smallest integer n such that any undirected graph G = (V; E) with n vertices must contain either a clique of size k or an independent set of size l. To give a more general formulation of Ramsey's Theorem, we rst dene the set theoretic (k; l) Ramsey property for a positive integer N . Ramsey Property N has the (k; l) Ramsey property if the following hol...
Generalizations of a RamseyTheoretic Result of Chv6tal
"... Chvátal has shown that if T is a tree on n points then r(Kk, T) _ (k 1) (n 1) + 1, where r is the (generalized) Ramsey number. It is shown that the same result holds when T is replaced by many other graphs. Such a T is called kgood. The results proved all support the conjecture that any large gr ..."
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Chvátal has shown that if T is a tree on n points then r(Kk, T) _ (k 1) (n 1) + 1, where r is the (generalized) Ramsey number. It is shown that the same result holds when T is replaced by many other graphs. Such a T is called kgood. The results proved all support the conjecture that any large graph that is sufficiently sparse, in the appropriate sense, is kgood. 1
On Some Multicolor Ramsey Numbers Involving K3 + e and K4 − e
"... Abstract: The Ramsey number R(G1, G2, G3) is the smallest positive integer n such that for all 3colorings of the edges of Kn there is a monochromatic G1 in the first color, G2 in the second color, or G3 in the third color. We study the bounds on various 3color Ramsey numbers R(G1, G2, G3), where G ..."
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Abstract: The Ramsey number R(G1, G2, G3) is the smallest positive integer n such that for all 3colorings of the edges of Kn there is a monochromatic G1 in the first color, G2 in the second color, or G3 in the third color. We study the bounds on various 3color Ramsey numbers R(G1, G2, G3), where Gi ∈ {K3, K3 + e, K4 − e, K4}. The minimal and maximal combinations of Gi’s correspond to the classical Ramsey numbers R3(K3) and R3(K4), respectively, where R3(G) = R(G, G, G). Here, we focus on the much less studied combinations between these two cases. Through computational and theoretical means we establish that R(K3, K3, K4 − e) = 17, and by construction we raise the lower bounds on R(K3, K4 − e, K4 − e) and R(K4, K4 − e, K4 − e). For some G and H it was known that R(K3, G, H) = R(K3 + e, G, H); we prove this is true for several more cases including R(K3, K3, K4 −e) = R(K3 +e, K3 +e, K4 −e). Ramsey numbers generalize to more colors, such as in the famous 4color case of R4(K3), where monochromatic triangles are avoided. It is known that 51 ≤ R4(K3) ≤ 62. We prove a surprising theorem stating that if R4(K3) = 51 then R4(K3 + e) = 52, otherwise R4(K3 + e) = R4(K3). 1
Ramsey Numbers Involving Cycles
"... We gather and review general results and data on Ramsey numbers involving cycles. This survey is based on the author’s 2009 revision #12 of the Dynamic Survey DS1, "Small Ramsey Numbers", at the ..."
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We gather and review general results and data on Ramsey numbers involving cycles. This survey is based on the author’s 2009 revision #12 of the Dynamic Survey DS1, "Small Ramsey Numbers", at the
Security in Computing ...
 MINDS AND MACHINES
, 1997
"... We prove that R(K5 −P3, K5) = 25 using computer algorithms. This solves one of the three remaining open cases in Hendry’s table, which listed the Ramsey numbers for pairs of graphs on 5 vertices. We find that there exist no (K5 −P3, K5)good graphs with a K4 on 23 or 24 vertices. The unique (K5 − P ..."
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We prove that R(K5 −P3, K5) = 25 using computer algorithms. This solves one of the three remaining open cases in Hendry’s table, which listed the Ramsey numbers for pairs of graphs on 5 vertices. We find that there exist no (K5 −P3, K5)good graphs with a K4 on 23 or 24 vertices. The unique (K5 − P3, K5)good graph with a K4 on 22 vertices is presented.
Computing the Ramsey Number R(K5 − P3, K5)
"... Abstract. We give a computerassisted proof of the fact that R(K5 − P3, K5) = 25. This solves one of the three remaining open cases in Hendry’s table, which listed the Ramsey numbers for pairs of graphs on 5 vertices. We find that there exist no (K5 −P3, K5)good graphs containing a K4 on 23 or 24 ..."
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Abstract. We give a computerassisted proof of the fact that R(K5 − P3, K5) = 25. This solves one of the three remaining open cases in Hendry’s table, which listed the Ramsey numbers for pairs of graphs on 5 vertices. We find that there exist no (K5 −P3, K5)good graphs containing a K4 on 23 or 24 vertices, where a graph F is (G, H)good if F does not contain G and the complement of F does not contain H. The unique (K5−P3, K5)good graph containing a K4 on 22 vertices is presented. 1