Hypergraphs with uniform Turán density equal to 8/27

Abstract

In the 1980s, Erdős and Sós initiated the study of Turán problems with a uniformity condition on the distribution of edges: the uniform Turán density of a hypergraph H is the infimum over all d for which any sufficiently large hypergraph with the property that all its linear-size subhypergraphs have density at least $d$ contains H. In particular, they asked to determine the uniform Turán densities of K4(3)- and K4(3). After more than 30 years, the former was solved in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter still remains open. Till today, there are known constructions of 3 -uniform hypergraphs with uniform Turán density equal to 0, 1/27, 4/27, and 1/4 only. We extend this list by a fifth value: we prove an easy to verify sufficient condition for the uniform Turán density to be equal to 8/27 and identify hypergraphs satisfying this condition. © The Author(s), 2025. Published by Cambridge University Press.