Hypergraphs with a Quarter Uniform Turán Density

Abstract

The uniform Turán density πu(F) of a (3-uniform) hypergraph F is the supremum of d for which there are infinitely many F-free hypergraphs with the property that every induced subhypergraph of H on a linearly sized vertex set has edge density at least d. Determining πu(F) for given hypergraphs F was suggested by Erdős and Sós in the 1980s. However, there are very few hypergraphs whose uniform Turán density has been determined. In this paper, we are the first to establish a verifiable condition for hypergraphs F with πu(F)=1/4. In particular, currently known hypergraphs whose uniform Turán density is 1/4, such as K4(3)- studied in Glebov et al. (Israel J Math 211:349–366, 2016) and Reiher et al. (J Eur Math Soc 20:1139–1159, 2018), and F5⋆ studied in Chen and Schülke (Beyond the broken tetrahedron, 2022, arXiv:2211.12747), satisfy this condition. Moreover, we also identify some new hypergraphs whose uniform Turán density is also 1/4.