Uniform Turán density -- palette classification

TL;DR

The paper advances the study of uniform Turán density for 3-uniform hypergraphs by reframing lower-bound constructions in terms of palette colorability and a Lagrangian $L({\cal P})$ of palettes. It proves a complete, verifiable criterion: the uniform Turán density of a hypergraph $H$ equals the supremum of $L({\cal P})$ over all palettes ${\cal P}$ for which $H$ is not ${\cal P}$-colorable, extending this to multiple palettes via a homomorphism-detection condition that involves symmetrization and products of palettes. This yields a powerful, constructive toolkit to realize specific densities (e.g., $4/81$) and to derive the existence of hypergraphs with prescribed uniform Turán densities without invoking heavy hypergraph regularity machinery. The results connect palette colorability to Turán-type extremal questions, offering both a conceptual simplification and practical methods for density realization and analysis.

Abstract

In the 1980s, Erdős and Sós initiated the study of Turán hypergraph problems with a uniformity condition on the distribution of edges, i.e., determining density thresholds for the existence of a hypergraph H in a host hypergraph with edges uniformly distributed. In particular, Erdős and Sós asked to determine the uniform Turán densities of the hypergraphs $K_4^{(3)-}$ and $K_4^{(3)}$. After more than 30 years, the former was solved by Glebov, Král' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, Rödl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter still remains open. In these two cases and several additional cases, the tight lower bounds are provided by a so-called palette construction. Lamaison [arXiv:2408.09643] has recently showed that the uniform Turán density of a 3-uniform hypergraph H is equal to the supremum of the densities of palettes that H is not colorable with. We give a necessary and sufficient condition, which is easy to verify, on the existence of a 3-uniform hypergraph colorable by a set of palettes and not colorable by another given set of palettes. We also demonstrate how our result can be used to prove the existence of 3-uniform hypergraphs with specific values of the uniform Turán density.

Figures (5)

• Figure 1: Visualization of the triples included in $T^{(\rm s)}$ for a triple $(x,y,z)\in T$.
• Figure 2: Visualization of a sought "subgrid" in the statement of Theorem \ref{['thm:grid']} for $n=3$ and $r=2$. The colors of pairs of points are depicted by (four) different types of lines.
• Figure 3: The feasible triples of the palette ${\cal P}_{\rm LM}$.
• Figure 4: The feasible triples of the palette ${\cal P}_{\rm 3T}$.
• Figure 5: The feasible triples of the palette ${\cal P}_{4/81}$.

Theorems & Definitions (35)

• Proposition 1
• Theorem 2
• Proposition 3
• Lemma 4
• Lemma 5
• Lemma 6
• proof
• Theorem 7: Erdős-Szekeres Theorem ErdS35
• Theorem 8: Ramsey's Theorem Ram30, multicolor hypergraph version
• Theorem 9