On zero-sum Ramsey numbers modulo 3

We start with a systematic study of the zero-sum Ramsey numbers. For a graph $G$ with $0(\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{m}\mathrm{o}\mathrm{d}3)$ edges, the zero-sum Ramsey number is defined as the smallest positive integer $R(G,{\mathbb{Z}}_3)$ such that for every $n\ge R(G,{\mathbb{Z}}_3)$ and every edge-colouring $f$ of $K_n$ using ${\mathbb{Z}}_3$, there is a zero-sum copy of $G$ in $K_n$ coloured by $f$, that is: ${\sum }_{e\in E(G)}f(e)\equiv 0(\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{m}\mathrm{o}\mathrm{d}3)$. Only sporadic results are known for these Ramsey numbers, and we discover many new ones. In particular we prove that for every forest $F$ on $n$ vertices and with $0(\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{m}\mathrm{o}\mathrm{d}3)$ edges, $R(F,{\mathbb{Z}}_3)\le n+2$, and this bound is tight if all the vertices of $F$ have degrees $1(\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{m}\mathrm{o}\mathrm{d}3)$. We also determine exact values of $R(T,{\mathbb{Z}}_3)$ for infinite families of trees.