On a problem of Caro on $\mathbb{Z}_3$-Ramsey number of forests

Abstract: Let $k$ be a positive integer and let $G$ be a graph. The zero-sum Ramsey number $R(G,\mathbb{Z}k)$ is the least integer $N$ (if it exists) such that for every edge-coloring $\chi \, : \, E(K_N) \, \rightarrow \, \mathbb{Z}_k$ one can find a copy of $G$ in $K_N$ such that $\sum{e \, \in \, E(G)}{\chi(e)} \, = \, 0$. In 2019, Caro made a conjecture about the $\mathbb{Z}_3$-Ramsey number of trees. In this paper, we settle this conjecture, fixing an incorrect case, and extend the result to forests. Namely, we show that \begin{equation*} R(F,\mathbb{Z}_3) = \left{ \begin{array}{ll} n+2, & \text{if $F$ is $1 (\mathrm{mod}\, 3)$ regular or a star;}\ n+1, & \text{if $3 \nmid d(v)$ for every $v \in V(F)$ or $F$ has exactly one} \ \phantom{placeholder} & \text{vertex of degree $0 (\mathrm{mod}\, 3)$ and all others are $1 (\mathrm{mod}\, 3)$,} \ \phantom{placeholder} & \text{and $F$ is not $1 (\mathrm{mod}\, 3)$ regular or a star;}\ n, & \text{otherwise.} \end{array} \right. \end{equation*} where $F$ is any forest on $n$ vertices with $3\mid e(F)$ and no isolated vertices.