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

For two given graphs G1and G2, the Ramsey number R (G1, G2) is the smallest integer n such that for any graph G of order n, either G contains G1or the complement of G contains G2. Let Cmdenote a cycle of length m and Kna 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.