A linear upper bound for zero-sum Ramsey numbers of bounded degree graphs

Abstract

Let $G$ be a graph and $\hat{I}“$ a finite abelian group. The zero-sum Ramsey number of $G$ over $\hat{I}“$, denoted by $R(G,\hat{I}“)$, is the smallest positive integer $t$ (if it exists) such that any edge-colouring $c:E(K_t)\to \hat{I}“$ contains a copy of $G$ with ${\sum }_{e\in E(G)}c(e)=0_{\hat{I}}“$. We prove a linear upper bound $R(G,\hat{I}“)\le Cn$ that holds for every $n$-vertex graph $G$ with bounded maximum degree and every finite abelian group $\hat{I}“$ with $|\hat{I}“|$ dividing $e(G)$.