Large monochromatic components in hypergraphs with large minimum codegree

TL;DR

This paper extends known results on large monochromatic components in hypergraphs, establishing optimal minimum codegree conditions for such components in k-uniform hypergraphs under any coloring.

Contribution

It proves the best possible minimum (k-1)-degree condition ensuring large monochromatic components in k-uniform hypergraphs, generalizing and strengthening previous graph and hypergraph results.

Findings

Monochromatic component size at least kn/(k+1) in hypergraphs

Optimal minimum (k-1)-degree bound established

Generalization of graph coloring results to hypergraphs

Abstract

A result of Gy\'arf\'as says that for every $3$-coloring of the edges of the complete graph $K_n$, there is a monochromatic component of order at least $\frac{n}{2}$, and this is best possible when $4$ divides $n$. Furthermore, for all $k\geq 3$ and every $(k+1)$-coloring of the edges of the complete $k$-uniform hypergraph $K_n^{k}$, there is a monochromatic component of order at least $\frac{kn}{k+1}$ and this is best possible for all $n$. Recently, Guggiari and Scott and independently Rahimi proved a strengthening of the graph case in the result above which says that the same conclusion holds if $K_n$ is replaced by any graph on $n$ vertices with minimum degree at least $\frac{5n}{6}-1$; furthermore, this bound on the minimum degree is best possible. We prove a strengthening of the $k\geq 3$ case in the result above which says that the same conclusion holds if $K_n^k$ is replaced by any $k$-uniform hypergraph on $n$ vertices with minimum $(k-1)$-degree at least $\frac{kn}{k+1}-(k-1)$; furthermore, this bound on the $(k-1)$-degree is best possible.