Coloring 2-intersecting hypergraphs

TL;DR

This paper proves that every 2-intersecting hypergraph can be colored with at most 5 colors to ensure each hyperedge has a sufficient number of distinct colors, answering a key question in hypergraph coloring.

Contribution

It establishes the optimal bound of 5 colors for coloring 2-intersecting hypergraphs, advancing understanding of hypergraph coloring constraints.

Findings

Every 2-intersecting hypergraph can be colored with at most 5 colors.

The bound of 5 colors is proven to be tight and optimal.

The result addresses a fundamental question in hypergraph coloring theory.

Abstract

A hypergraph is 2-intersecting if any two edges intersect in at least two vertices. Blais, Weinstein and Yoshida asked (as a first step to a more general problem) whether every 2-intersecting hypergraph has a vertex coloring with a constant number of colors so that each hyperedge e has at least min{|e|,3} colors. We show that there is such a coloring with at most 5 colors (which is best possible).