On three open problems in zero-sum Ramsey numbers

Abstract

Let $K_N^{(r)}$ denote the $N$-vertex complete $r$-uniform hypergraph. For an $r$-uniform hypergraph $H$ and an integer $k\ge 2$, the $k$-color Ramsey number $R(H,k)$ is the least integer $N$ such that every $k$-edge-coloring of $K_N^{(r)}$ contains a monochromatic copy of $H$. When $k\mid\esize(H)$, the zero-sum Ramsey number $R(H,{\mathbb{Z}}_k)$ is the least integer $N$ such that every edge-labeling of $K_N^{(r)}$ by elements of ${\mathbb{Z}}_k$ contains a copy of $H$ whose edge labels sum to $0$ in ${\mathbb{Z}}_k$. We settle two conjectures and a problem concerning these two Ramsey numbers. First, Caro and Provstgaard proposed exact values for the zero-sum Ramsey numbers over ${\mathbb{Z}}_2$ of delta-systems with an even number of edges. We determine these numbers and thereby prove their conjecture. Second, for a forest $F$ with $m$ edges, let $tF$ denote the disjoint union of $t$ copies of $F$. Caro conjectured that $R(tF,{\mathbb{Z}}_{mt})=R(tF,2)$ for all sufficiently large $t$. We show that this conjecture does not hold for double stars. Caro also asked whether there exists a tree $T$ with $m$ edges such that $R(T,{\mathbb{Z}}_m)>R(T,2)$. We answer this question affirmatively by constructing an infinite family of such trees.