A linear upper bound on the zero-sum Ramsey number of forests in ${\mathbb{Z}}_p$

Abstract

Let $m$ be a positive integer and let $G$ be a graph. The zero-sum Ramsey number $R(G,{\mathbb{Z}}_m)$ is the least integer $N$ (if it exists) such that for every edge-coloring $\chi :E(K_N)\to {\mathbb{Z}}_m$ one can find a copy of $G$ in $K_N$ such that ${\sum }_{e\in E(G)}\chi (e)=0$. In this paper, we show that, for every prime $p$, $$R(F,{\mathbb{Z}}_p)\le n+9p-12$$ for every forest $F$ in $n\ge 3p^2-12p+11$ vertices with $p\mid e(F)$ without isolated vertices.