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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...
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...