The Ramsey numbers for a cycle of length six or seven versus a clique of order seven

Abstract

For two given graphs G₁ and G₂, the Ramsey number R(G₁,G₂) is the smallest integer n such that for any graph G of order n, either G contains G₁ or the complement of G contains G₂. Let C[sub m] denote a cycle of length m and K[sub n] a complete graph of order n. It was conjectured that R(C[sub m],K[sub n])=(m-1)(n-1)+1 for m≥n≥3 and (m,n)≠(3,3). We show that R(C[sub 6],K[sub 7])=31 and R(C[sub 7],K[sub 7])=37, and the latter result confirms the conjecture in the case when m=n=7.