Large monochromatic components in colorings of complete hypergraphs

TL;DR

This paper extends classical results on monochromatic components from complete graphs to hypergraphs, providing tight bounds and analyzing the structure of large monochromatic components across multiple colors.

Contribution

It generalizes the concept of monochromatic connected components to hypergraphs and offers tight bounds for their sizes in multi-colorings, including detailed analysis for two-color cases.

Findings

Established tight bounds for monochromatic component sizes in hypergraphs.

Extended connectivity concepts to higher uniformities with bounds close to optimal.

Provided precise results for two-color hypergraph colorings.

Abstract

Gy\'arf\'as famously showed that in every $r$-coloring of the edges of the complete graph $K_n$, there is a monochromatic connected component with at least $\frac{n}{r-1}$ vertices. A recent line of study by Conlon, Tyomkyn, and the second author addresses the analogous question about monochromatic connected components with many edges. In this paper, we study a generalization of these questions for $k$-uniform hypergraphs. Over a wide range of extensions of the definition of connectivity to higher uniformities, we provide both upper and lower bounds for the size of the largest monochromatic component that are tight up to a factor of $1+o(1)$ as the number of colors grows. We further generalize these questions to ask about counts of vertex $s$-sets contained within the edges of large monochromatic components. We conclude with more precise results in the particular case of two colors.