Hypergraphs with uniform Turán density equal to 8/27
Frederik Garbe, Daniel Iľkovič, Daniel Kráľ, Filip Kučerák, Ander Lamaison
TL;DR
The paper addresses the problem of determining the uniform Turán density for 3-uniform hypergraphs, focusing on augmenting the known density values by introducing a palette-based criterion. It proves an easy-to-verify sufficient condition: if a hypergraph H is Φ3-colorable but not Φ8-colorable, then its uniform Turán density is exactly $8/27$, and it provides explicit hypergraphs meeting this condition. The authors establish an embedding upper bound via partitioned-hypergraph machinery and construct Φ3-colorable but Φ8-uncolorable examples to realize the $8/27$ threshold, including a concrete instance from ${ m 𝔽}_5^5$. The work strengthens the palette-based approach as a powerful tool for classifying uniform Turán densities and points toward a complete description via palette noncolorability, with further connections to recent developments by Lam.
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.