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

TL;DR

This paper establishes a linear upper bound on zero-sum Ramsey numbers for bounded degree graphs over finite abelian groups, advancing understanding of combinatorial coloring problems.

Contribution

It proves a linear upper bound for zero-sum Ramsey numbers of bounded degree graphs over finite abelian groups, generalizing previous results.

Findings

Zero-sum Ramsey number $R(G, \Gamma)$ is bounded above by a linear function of the number of vertices.

The bound applies to all graphs with bounded maximum degree.

The result holds for all finite abelian groups where the group order divides the number of edges in the graph.

Abstract

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