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 $χ\, : \, E(K_N) \, \rightarrow \, \mathbb{Z}_m$ one can find a copy of $G$ in $K_N$ such that $\sum{e \, \in \, E(G)}{χ(e)} \, = \, 0$. In this paper, we show that, for every prime $p$, $$R(F,\mathbb{Z}_p)\leq n+9p-12$$ for every forest $F$ in $n\geq 3p^2-12p+11$ vertices with $p\mid e(F)$.