Hypergraphs of Arbitrary Uniformity with Vanishing Codegree Turán Density

Abstract.

The codegree Turán density [Formula: see text] of a [Formula: see text]-uniform hypergraph (or [Formula: see text]-graph) [Formula: see text] is the infimum over all [Formula: see text] such that a copy of [Formula: see text] is contained in any sufficiently large [Formula: see text]-vertex [Formula: see text]-graph [Formula: see text] with the property that any [Formula: see text]-subset of [Formula: see text] is contained in at least [Formula: see text] edges. The problem of determining [Formula: see text] for a [Formula: see text]-graph [Formula: see text] is in general very difficult when [Formula: see text], and there were previously very few nontrivial examples of [Formula: see text]-graphs [Formula: see text] for which [Formula: see text] was known when [Formula: see text]. In this paper, we prove that [Formula: see text], the [Formula: see text]-uniform tight cycle of length [Formula: see text] minus an edge, has vanishing codegree Turán density if and only if [Formula: see text] (mod [Formula: see text]) when [Formula: see text]. This generalizes a result of Piga, Sales, and Schülke, who proved that [Formula: see text] when [Formula: see text]. The method used to prove that [Formula: see text] when [Formula: see text] (mod [Formula: see text]) and [Formula: see text] in fact gives a rather larger class of [Formula: see text]-graphs with vanishing codegree Turán density. We also answer a question of Piga and Schülke by proving that another family of [Formula: see text]-graphs, studied by them, has vanishing codegree Turán density.