The optimal linear bound for zero-sum Ramsey numbers of forests

Let $p$ be prime, let $n_p=n(p)$ be a positive integer, and let $F$ be a forest on $n\geq n_p$ vertices such that $p\mid e(F)$. Here $R(F,\mathbb{Z}_p)$ denotes the least order of a complete graph such that every edge-coloring by $\mathbb{Z}_p$ contains a zero-sum copy of $F$. The optimal forest bound. For every prime $p$, there is a positive integer $n_p=n(p)$ such that

$$R(F,\mathbb{Z}_p)\leq n+p-1.$$

The paper proves the weaker bound $R(F,\mathbb{Z}_p)\leq n+9p-12$ for forests with $n\geq 3p^2-12p+11$, while constructions show that the additive constant cannot be smaller than $p-1$.