Counting induced subgraphs with given intersection sizes

Let $F$ be a graph of order $r$. In this paper, we study the maximum number of induced copies of $F$ with restricted intersections, which highlights the motivation from extremal set theory. Let $L=\{\ell_1,\dots,\ell_s\}\subseteq[0,r-1]$ be an integer set with $s\not\in\{1,r\}$. Let $\Psi_r(n,F,L)$ be the maximum number of induced copies of $F$ in an $n$-vertex graph, where the induced copies of $F$ are $L$-intersecting as a family of $r$-subsets, i.e., for any two induced copies of $F$, the size of their intersection is in $L$. Helliar and Liu initiated a study of the function $\Psi_r(n,K_r,L)$. Very recently, Zhao and Zhang improved their result and showed that $\Psi_r(n,K_r,L)=\Theta_{r,L}(n^{s})$ if and only if $\ell_1,\dots,\ell_s,r$ form an arithmetic progression. In this paper, we show that $\Psi_r(n,F,L)=o_{r,L}(n^{s})$ when $\ell_1,\dots,\ell_s,r$ do not form an arithmetic progression. We study the asymptotical result of $\Psi_r(n,C_r,L)$, and determined the asymptotically optimal result when $\ell_1,\dots,\ell_s,r$ form an arithmetic progression and take certain values. We also study the generalized Tur\'an problem, determining the maximum number of $H$, where the copies of $H$ are $L$-intersecting as a family of $r$-subsets. The entropy method is used to prove our results.