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 $K_4^{(3)-}$ and $K_4^{(3)}$. 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 condition for the uniform Turán density to be equal to 8/27 and identify hypergraphs satisfying this condition.

Minor revision. Small corrections and improvements to the text