On a problem concerning ${\mathbb{Z}}_3$-Ramsey numbers

Abstract

For a graph $G$ with $3\mid e(G)$, let $R(G,{\mathbb{Z}}_3)$ denote the least integer $N$ such that every edge weighting $w:E(K_N)\to {\mathbb{Z}}_3$ admits a copy of $G$ whose edge weights sum to zero. We prove that every $n$-vertex graph $G$ with $n\ge 6$ and $3\mid e(G)$ satisfies $R(G,{\mathbb{Z}}_3)\le n+5$. Moreover, the stronger bound $R(G,{\mathbb{Z}}_3)\le n+4$ holds whenever $G$ contains an induced $P_4$ or an induced $2K_2$. Consequently, we resolve a problem posed by Caro and Mifsud: every graph $G$ with $3\mid e(G)$ satisfies $R(G,{\mathbb{Z}}_3)\le |V(G)|+8$, with equality if and only if $G\cong K_3$.