Hypergraphs with uniform Tur\'an density equal to 8/27

TL;DR

This paper establishes a new result in hypergraph Turán theory by identifying conditions under which the uniform Turán density equals 8/27, expanding the known set of possible densities.

Contribution

The authors provide a simple criterion for hypergraphs to have uniform Turán density 8/27 and identify hypergraphs satisfying this condition, adding a new value to the known densities.

Findings

Proved a condition for uniform Turán density to be 8/27.

Identified hypergraphs satisfying this condition.

Expanded the set of known uniform Turán densities.

Abstract

In the 1980s, Erd\H{o}s and S\'os initiated the study of Tur\'an problems with a uniformity condition on the distribution of edges: the uniform Tur\'an 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 Tur\'an 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 Tur\'an 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 Tur\'an density to be equal to 8/27 and identify hypergraphs satisfying this condition.