Hypergraphs with Zero Chromatic Threshold

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 (|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 Erdős-Simonovits. One interesting question, first proposed by Luczak-Thomassé and then solved by Allen-BöttcherGriffiths-Kohayakawa-Morris, is the characterization of graphs having zero chromatic threshold, in particular the fact that there are graphs with non-zero Turán 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 Bollobás-Erdős graph defined earlier by the authors.