New Computational Upper Bounds for Ramsey Numbers $R(3,k)$
Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: $R(3,10) \le 42$, $R(3,11) \le 50$, $R(3,13) \le 68$, $R(3,14) \le 77$, $R(3,15) \le 87$, and $R(3,16) \le 98$. All of them are improvements by one over the previously best known 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 $k \leq 9$ and for all $n \leq 33$ for $k = 10$, and by establishing new lower bounds on $e(3,k,n)$ for most of the open cases for $10 \le k \le 15$. The enumeration of all graphs witnessing the values of $e(3,k,n)$ is completed for all cases with $k \le 9$. 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 $R(3,10) \le 42$.