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...

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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...

For a graph G, the expression G v → (a1,..., ar) means that for any r-coloring of the vertices of G there exists a monochromatic ai-clique in G for some color i ∈ {1,..., r}. The vertex Folkman numbers are defined as Fv(a1,..., ar; q) = min{|V (G) | : G v → (a1,..., ar) and Kq * G}. Of these, the only Folkman number of the form F (2,..., 2; r − 1) which has remained unknown up to this r time is Fv(2, 2, 2, 2, 2; 4). We show here that Fv(2, 2, 2, 2, 2; 4) = 16, which is equivalent to saying that the smallest 6-chromatic K4-free graph has 16 vertices. We also show that the sole witnesses of the upper bound Fv(2, 2, 2, 2, 2; 4) ≤ 16 are the two Ramsey (4,4)-graphs on 16 vertices. 1