A linear upper bound on zero-sum Ramsey numbers of graphs

Let $p$ be prime, let $c_p=c(p)$ be an integer, and let $G$ be a graph on $n$ vertices such that $p\mid E(G)$. The graph linear-bound conjecture. There is an integer $c_p=c(p)$ such that, for all $n$ and all graphs $G$ on $n$ vertices,

$$R(G,\mathbb{Z}_p)\leq n+c_p.$$

The paper establishes a linear bound for forests and proposes this as a generalization to arbitrary graphs. The conjecture asks whether the additive constant can depend only on the prime $p$, uniformly over all graph orders and all graphs satisfying the divisibility condition.