On the Zero-Sum Ramsey Problem over $\mathbb {Z}_2^d$

Abstract

Let $\varGamma$ be a finite abelian group and let G be a graph. The zero-sum Ramsey number $R(G, \varGamma )$ is the least integer N (if it exists) such that, for every edge-colouring $E(K_N) \mapsto \varGamma$, one can find a copy $G \subseteq K_N$ where the sum of the colours of the edges of G is zero.

A large body of research on this problem has emerged in the past few decades, paying special attention to the case where $\varGamma$ is cyclic. In this work, we start a systematic study of $R(G, \varGamma )$ for groups $\varGamma$ of small exponent, in particular, $\varGamma = \mathbb {Z}_2^d$. For the Klein group $\mathbb {Z}_2^2$, the smallest non-cyclic abelian group, we compute $R(G, \mathbb {Z}_2^2)$ for all odd forests G and show that $R(G, \mathbb {Z}_2^2) \le n+2$ for all forests G on at least 6 vertices. We also show that $R(C_4, \mathbb {Z}_2^d) = 2^d + 1$ for any $d \ge 2$, and determine the order of magnitude of $R(K_{t,r}, \mathbb {Z}_2^d)$ as $d \rightarrow \infty$ for all t, r.

We also consider the related setting where the ambient graph $K_N$ is substituted by the complete bipartite graph $K_{N,N}$. Denote the analogue bipartite zero-sum Ramsey number by $B(G, \varGamma )$. We show that $B(C_4, \mathbb {Z}_2^d) = 2^d + 1$ for all $d \ge 1$ and $B(\{C_4, C_6\}, \mathbb {Z}_2^d) = 2^{d/2} + 1$ for all even $d \ge 2$. Additionally, we show that $B(K_{t,r}, \mathbb {Z}_2^d)$ and $R(K_{t,r}, \mathbb {Z}_2^d)$ have the same asymptotic behaviour as $d \rightarrow \infty$, for all t, r. Finally, we conjecture the value of $B(\{C_4, \dotsc , C_{2m}\}, \mathbb {Z}_2^d)$ and provide the corresponding lower bound.