A local clique density theorem in $H$-free graphs

Abstract

In 2016, Reiher's clique density theorem determined the minimum number of copies of $K_t$ in a graph with a prescribed edge density. In this paper, we investigate its local version and prove a local clique density theorem in $H$-free graphs as follows. For integers $r$ and $t$ with $2\le t\le r-1$, any $r$-chromatic graph $H$, any real numbers $\gamma$ and $\alpha$ with $\frac{t-2}{2(t-1)}\le \gamma \le \frac{r-2}{2(r-1)}$ and $0\le \alpha \le 1$, we determine the maximum value $\beta :=\beta (r,t,\alpha ,\gamma )$ such that for every $n$-vertex $H$-free graph $G$ with at least $\gamma n^2$ edges, every $\lceil \alpha n\rceil$-vertex subset in $G$ contains at least $(\beta -o(1))n^t$ copies of $K_t$. In particular, when $H=K_r$, every $\lceil \alpha n\rceil$-vertex subset contains at least $\lfloor \beta n^t\rfloor$ copies of $K_t$, which is an exact bound. For suitable choices of $\alpha$ and $\gamma$, namely, those for which all part ratios in the corresponding extremal construction are rational, this bound is attained for infinitely many values of $n$.