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arXiv · 2609.27788

On a problem concerning $\mathbb Z_3$-Ramsey numbers

Cheng Chi·Jialin He
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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 $n\ge6$ 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 $G\cong K_3$.

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Cheng Chi, Jialin He. 2026-08-17. On a problem concerning $\mathbb Z_3$-Ramsey numbers. https://arxiv.org/abs/2609.27788

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