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

AbstractFor 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 Cm denote a cycle of length m and Kn a complete graph of order n. It was conjectured that R(Cm,Kn)=(m-1)(n-1)+1 for m⩾n⩾3 and (m,n)≠(3,3). We show that R(C6,K7)=31 and R(C7,K7)=37, and the latter result confirms the conjecture in the case when m=n=7