Codegree Turán density of complete $r$-uniform hypergraphs

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 Turán 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 \[ 1 - c_2 \frac{\ln t}{t^{r-1}} \le γ(K_t^r) \le 1 - c_1 \frac{\ln t}{t^{r-1}}, \] where $K_t^r$ is the complete $r$-graph on $t$ vertices. This gives the best general bounds for $\gamma (K_t^r)$.

arXiv commentfixed typos, accepted for publication in SIAM J. Discrete Math