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

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 $\gamma (K_t^r)$.