Hypergraphs with Zero Chromatic Threshold

TL;DR

This paper investigates the chromatic thresholds of r-uniform hypergraphs, identifying conditions under which these thresholds are zero or non-zero, and introduces new constructions based on a special product of the Bollobás-Erdős graph.

Contribution

It advances understanding of hypergraph chromatic thresholds by characterizing classes with zero thresholds and providing new constructions for non-zero thresholds.

Findings

Large classes of hypergraphs have zero chromatic threshold.

Constructed hypergraphs with non-zero chromatic threshold.

Used a special product of Bollobás-Erdős graphs for constructions.

Abstract

Let F be an r-uniform hypergraph. The chromatic threshold of the family of F-free, r-uniform hypergraphs is the infimum of all non-negative reals c such that the subfamily of F-free, r-uniform hypergraphs H with minimum degree at least $c \binom{|V(H)|}{r-1}$ has bounded chromatic number. The study of chromatic thresholds of various graphs has a long history, beginning with the early work of Erd\H{o}s-Simonovits. One interesting question, first proposed by \L{}uczak-Thomass\'{e} and then solved by Allen-B\"{o}ttcher-Griffiths-Kohayakawa-Morris, is the characterization of graphs having zero chromatic threshold, in particular the fact that there are graphs with non-zero Tur\'{a}n density that have zero chromatic threshold. In this paper, we make progress on this problem for r-uniform hypergraphs, showing that a large class of hypergraphs have zero chromatic threshold in addition to exhibiting a family of constructions showing another large class of hypergraphs have non-zero chromatic threshold. Our construction is based on a special product of the Bollob\'as-Erd\H{o}s graph defined earlier by the authors.