The Ramsey Number R(3, K10-e) and Computational Bounds for R(3, G)

Using computer algorithms we establish that the Ramsey number R(3,K10− e) is equal to 37, which solves the smallest open case for Ramsey numbers of this type. We also obtain new upper bounds for the cases of R(3,Kk − e) for 11 ≤ k ≤ 16, and show by construction a new lower bound 55 ≤ R(3,K13 − e). The new upper bounds on R(3,Kk − e) are obtained by using the values and lower bounds on e(3,Kl − e, n) for l ≤ k, where e(3,Kk − e, n) is the minimum number of edges in any triangle-free graph on n vertices without Kk − e in the complement. We complete the computation of the exact values of e(3,Kk − e, n) for all n with k ≤ 10 and for n ≤ 34 with k = 11, and establish many new lower bounds on e(3,Kk − e, n) for higher values of k. Using the maximum triangle-free graph generation method, we determine two other previously unknown Ramsey numbers, namely R(3,K10 −K3 − e) = 31 and R(3,K10 − P3 − e) = 31. For graphs G on 10 vertices, besides G = K10, this leaves 6 open cases of the form R(3, G). The hardest among them appears to be G = K10 − 2K2, for which we establish the bounds 31 ≤ R(3,K10 − 2K2) ≤ 33.