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 $c\left(\genfrac{}{}{0px}{}{\mathrm{\mid }V(H)\mathrm{\mid }}{r-1}\right)$ 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.