Large monochromatic components in expansive hypergraphs

Abstract

A result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary $r$-colouring of the complete $k$-uniform hypergraph $K_n^k$ when $k\geq 2$ and $k\in \{r-1,r\}$. We prove a result which says that if one replaces $K_n^k$ in Gyárfás’ theorem by any ‘expansive’ $k$-uniform hypergraph on $n$ vertices (that is, a $k$-uniform hypergraph $G$ on $n$ vertices in which $e(V_1, \ldots, V_k)\gt 0$ for all disjoint sets $V_1, \ldots, V_k\subseteq V(G)$ with $|V_i|\gt \alpha$ for all $i\in [k]$), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on $r$ and $\alpha$). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms.

Gyárfás’ result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary $r$-partite $r$-uniform hypergraph $H$ with $n$ edges in which every set of $k$ edges has a common intersection. In this language, our result says that if one replaces the condition that every set of $k$ edges has a common intersection with the condition that for every collection of $k$ disjoint sets $E_1, \ldots, E_k\subseteq E(H)$ with $|E_i|\gt \alpha$, there exists $(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$ such that $e_1\cap \cdots \cap e_k\neq \emptyset$, then the smallest possible maximum degree of $H$ is essentially the same (within a small error term depending on $r$ and $\alpha$). We prove our results in this dual setting.