paper

Extremal problems in uniformly dense hypergraphs and digraphs

arXiv:2603.10766

Abstract

The uniform TurÃ¡n density $I ¨ _{u} (F)$ of a $3$ -uniform hypergraph (or $3$ -graph) $F$ is the supremum of all $d$ such that there exist infinitely many $F$ -free $3$ -graphs $H$ in which every induced subhypergraph on a linearly sized vertex set has edge density at least $d$ . Determining $I ¨ _{u} (F)$ for a given $3$ -graph $F$ was proposed by ErdÅs and SÃ³s in the 1980s, yet only a few cases are known. In particular, it remains open whether $1/2$ can occur as a value of $I ¨ _{u}$ . In this paper, we establish a novel connection between TurÃ¡n-type extremal problems for digraphs and uniform TurÃ¡n densities of $3$ -graphs. Using digraph extremal results, we give the first verifiable conditions for $3$ -graphs $F$ with $I ¨ _{u} (F) = (r - 1) / r$ and $I ¨ _{u} (F) = (r - 1)^{2} / r^{2}$ for all $r \geq 2$ , and identify the corresponding $3$ -graphs. In particular, these $3$ -graph classes contain some specific $3$ -graphs, such as $K_{4}^{(3) -}$ . We also present a sufficient condition ensuring $I ¨ _{u} (F) = 4/27$ and construct $3$ -graphs satisfying it; in particular, our examples are different from the tight $3$ -uniform cycles whose uniform TurÃ¡n density $4/27$ was determined in [{Trans. Amer. Math. Soc. 376 (2023), 4765-4809}]. Finally, we give a short proof of the existence of $3$ -graphs $F$ with $I ¨ _{u} (F) = 1/27$ , originally established by Garbe, KrÃ¡l' and Lamaison [{Israel J. Math. 259 (2024), 701-726}] via the hypergraph regularity method.

20 pages, 2 figures