New Computational Upper Bounds for Ramsey Numbers R(3, k)

Using computational techniques we derive six new upper bounds on the classical twocolor Ramsey numbers: R(3, 10) ≤ 42, R(3, 11) ≤ 50, R(3, 13) ≤ 68, R(3, 14) ≤ 77, R(3, 15) ≤ 87, and R(3, 16) ≤ 98. All of them are improvements by one over the previously best published bounds. Let e(3, k, n) denote the minimum number of edges in any triangle-free graph on n vertices without independent sets of order k. The new upper bounds on R(3, k) are obtained by completing the computation of the exact values of e(3, k, n) for all n with k ≤ 9 and for all n ≤ 33 for k = 10, and by establishing new lower bounds on e(3, k, n) for most of the open cases for 10 ≤ k ≤ 15. The enumeration of all graphs witnessing the values of e(3, k, n) is completed for all cases with k ≤ 9. We prove that the known critical graph for R(3, 9) on 35 vertices is unique up to isomorphism. For the case of R(3, 10), first we establish that R(3, 10) = 43 if and only if e(3, 10, 42) = 189, or equivalently, that if R(3, 10) = 43 then every critical graph is regular of degree 9. Then, using computations, we disprove the existence of the latter, and thus show that R(3, 10) ≤ 42.