Coloring triple systems with local conditions

TL;DR

This paper constructs a specialized edge-coloring of complete 3-uniform hypergraphs ensuring local diversity in colors for every five vertices, addressing a key open problem in hypergraph coloring.

Contribution

It provides the first known coloring with sub-polynomial number of colors satisfying a local color diversity condition in 3-uniform hypergraphs.

Findings

Established a coloring with $e^{O( oot{log log n})}$ colors

Resolved the first open case of the Conlon-Fox-Lee-Sudakov question

Demonstrated existence of colorings with fewer than polynomial colors

Abstract

We produce an edge-coloring of the complete 3-uniform hypergraph on n vertices with $e^{O(\sqrt {log log n})}$ colors such that the edges spanned by every set of five vertices receive at least three distinct colors. This answers the first open case of a question of Conlon-Fox-Lee-Sudakov [1] who asked whether such a coloring exists with $(log n)^{o(1)}$ colors.