$R(K_6-e,K_4)=30$

Abstract

We settle the Ramsey problem $R(K_6-e,K_4)$, also known as $R(J_6,K_4)$ and $R(K_6^{-},K_4)$. Previously, the best bounds were $30\le R(K_6-e,K4)\le 32$. We prove that $R(K_6-e,K_4)=30$. Our technique is based on the recent approach of Angeltveit and McKay and on older algorithms of McKay and Radziszowski.