On local Turán density problems of hypergraphs

Chunqiu Fang; Guorong Gao; Jie Ma; Ge Song

paper

On local Turán density problems of hypergraphs

arXiv:2303.00427

Abstract

For integers $q \geq p \geq r \geq 2$ , we say that an $r$ -uniform hypergraph $H$ has property $(q, p)$ , if for any $q$ -vertex subset $Q$ of $V (H)$ , there exists a $p$ -vertex subset $P$ of $Q$ spanning a clique in $H$ . Let $T_{r} (n, q, p) = min {e (H) : H \subset (r [ n ]), H has property (q, p)}$ . The local Turán density about property $(q, p)$ in $r$ -uniform hypergraphs is defined as $t_{r} (q, p) = lim_{n \to \infty} T_{r} (n, q, p) / (r n)$ . Frankl, Huang and Rödl [J. Comb. Theory, Ser. A, 177 (2021)] showed that $lim_{p \to \infty} t_{r} (a p + 1, p + 1) = \frac{1}{a ^{r - 1}}$ for positive integer $a$ and $t_{3} (2 p + 1, p + 1) = \frac{1}{4}$ for all $p \geq 3$ and asked the question that determining the value of $lim_{p \to \infty} t_{r} (γ p + 1, p + 1)$ , where $γ \geq 1$ is a real number. Based on the study of hypergraph Turán densities, we determine some exact values of local Turán densities and answer their question partially; in particular, our results imply that the equality in their question about exact values does not hold in general.