On 3‐graphs with vanishing codegree Turán density

Source abstract

Abstract For a ‐uniform hypergraph (or simply ‐graph) , the codegree Turán density is the supremum over all such that there exist arbitrarily large ‐vertex ‐free ‐graphs in which every ‐subset of is contained in at least edges. In this paper, we study the problem of what 3‐graphs satisfy . We find that this is closely related to the uniform Turán density , which is the supremum over all such that there are infinitely many ‐free ‐graphs satisfying that any induced linear‐size subhypergraph of has edge density at least . We prove that, for every 3‐graph , implies . We also introduce a layered structure for 3‐graphs which allows us to obtain the reverse implication: Every layered 3‐graph with satisfies . Along the way, we answer in the negative a question of Falgas‐Ravry, Pikhurko, Vaughan, and Volec [J. Lond. Math. Soc. (2) 107 (2023), 1660–1691] about whether always holds. In particular, we construct counterexamples with positive but arbitrarily small while having . Our proof relies on a random geometric construction, graph distributions, Ramsey's theorem and a new formulation of the characterization of 3‐graphs with vanishing uniform Turán density due to Reiher, Rödl, and Schacht [J. Lond. Math. Soc. 97 (2018), no. 1, 77–97].