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

In the 1980s, Erd{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 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 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.