Large almost monochromatic subsets in hypergraphs

TL;DR

This paper proves that in any coloring of triples of an N-element set with  or more colors, there exists a large subset where almost all triples are the same color, contrasting with the much smaller monochromatic subsets.

Contribution

It establishes a tight bound on the size of almost monochromatic subsets in hypergraph colorings, answering longstanding open questions by Erd
51s and Hajnal.

Findings

Existence of large almost monochromatic subsets in hypergraphs

New upper bounds on -color Ramsey numbers for hypergraphs

Contrast between almost monochromatic and monochromatic subset sizes

Abstract

We show that for all $\ell$ and $\epsilon>0$ there is a constant $c=c(\ell,\epsilon)>0$ such that every $\ell$-coloring of the triples of an $N$-element set contains a subset $S$ of size $c\sqrt{\log N}$ such that at least $1-\epsilon$ fraction of the triples of $S$ have the same color. This result is tight up to the constant $c$ and answers an open question of Erd\H{o}s and Hajnal from 1989 on discrepancy in hypergraphs. For $\ell \geq 4$ colors, it is known that there is an $\ell$-coloring of the triples of an $N$-element set whose largest monochromatic subset has cardinality only $\Theta(\log \log N)$. Thus, our result demonstrates that the maximum almost monochromatic subset that an $\ell$-coloring of the triples must contain is much larger than the corresponding monochromatic subset. This is in striking contrast with graphs, where these two quantities have the same order of magnitude. To prove our result, we obtain a new upper bound on the $\ell$-color Ramsey numbers of complete multipartite 3-uniform hypergraphs, which answers another open question of Erd\H{o}s and Hajnal.