The Ramsey Number $R(3,K_{10}-e)$ and Computational Bounds for $R(3,G)$

Abstract

Using computer algorithms we establish that the Ramsey number $R(3,K_{10}-e)$ 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 $R(3,K_k-e)$ for $11 \le k \le 16$, and show by construction a new lower bound $55 \le R(3,K_{13}-e)$. The new upper bounds on $R(3,K_k-e)$ are obtained by using the values and lower bounds on $e(3,K_l-e,n)$ for $l \le k$, where $e(3,K_k-e,n)$ is the minimum number of edges in any triangle-free graph on $n$ vertices without $K_k-e$ in the complement. We complete the computation of the exact values of $e(3,K_k-e,n)$ for all $n$ with $k \leq 10$ and for $n \leq 34$ with $k = 11$, and establish many new lower bounds on $e(3,K_k-e,n)$ for higher values of $k$. Using the maximum triangle-free graph generation method, we determine two other previously unknown Ramsey numbers, namely $R(3,K_{10}-K_3-e)=31$ and $R(3,K_{10}-P_3-e)=31$. For graphs $G$ on 10 vertices, %besides $G=K_{10}$, this leaves 6 other open besides $G=K_{10}$, this leaves 6 open cases of the form $R(3,G)$. The hardest among them appears to be $G=K_{10}-2K_2$, for which we establish the bounds $31 \le R(3,K_{10}-2K_2) \le 33$.