Codegree Turan density of complete $r$-uniform hypergraphs

Let $rge 3$. 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 Turan 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 $gamma(K_t^r)$.