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

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 $\chi :E(K_N)\to {\mathbb{Z}}_k$ 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, & $\text{if }F\text{ is }1(\mathrm{m}\mathrm{o}\mathrm{d}3)\text{ regular or a star;}$\ n+1, & $\text{if }3\nmid d(v)\text{ for every }v\in V(F)\text{ or }F\text{ has exactly one}$ \ \phantom{placeholder} & $\text{vertex of degree }0(\mathrm{m}\mathrm{o}\mathrm{d}3)\text{ and all others are }1(\mathrm{m}\mathrm{o}\mathrm{d}3)\text{,}$ \ \phantom{placeholder} & $\text{and }F\text{ is not }1(\mathrm{m}\mathrm{o}\mathrm{d}3)\text{ regular or a star;}$\ 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.