Codegree Tur\'an density of complete $r$-uniform hypergraphs

TL;DR

This paper establishes tight bounds on the codegree Turán density of complete r-uniform hypergraphs, revealing how minimum codegree conditions influence the presence of such subhypergraphs.

Contribution

It provides the first precise bounds on the codegree Turán density for complete r-uniform hypergraphs, improving understanding of hypergraph extremal problems.

Findings

Bounds depend on logarithmic factors of t

Constants c1 and c2 depend only on r

Results are the best general bounds known

Abstract

Let $r\ge 3$. Given an $r$-graph $H$, the minimum codegree $\delta_{r-1}(H)$ is the largest integer $t$ such that every $(r-1)$-subset of $V(H)$ is contained in at least $t$ edges of $H$. Given an $r$-graph $F$, the codegree Tur\'an density $\gamma(F)$ is the smallest $\gamma >0$ such that every $r$-graph on $n$ vertices with $\delta_{r-1}(H)\ge (\gamma + o(1))n$ contains $F$ as a subhypergraph. Using results on the independence number of hypergraphs, we show that there are constants $c_1, c_2>0$ depending only on $r$ such that \[ 1 - c_2 \frac{\ln t}{t^{r-1}} \le \gamma(K_t^r) \le 1 - c_1 \frac{\ln t}{t^{r-1}}, \] where $K_t^r$ is the complete $r$-graph on $t$ vertices. This gives the best general bounds for $\gamma(K_t^r)$.