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=\₁, , ₛ\0, r-1 be an integer set with s\1, r\. Let Ψᵣ (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 Ψᵣ (n, Kᵣ, L). Very recently, Zhao and Zhang improved their result and showed that Ψᵣ (n, Kᵣ, L) =Θₑ, ₋ (n^s) if and only if ₁, , ₛ, r form an arithmetic progression. In this paper, we show that Ψᵣ (n, F, L) =oₑ, ₋ (n^s) when ₁, , ₛ, r do not form an arithmetic progression. We study the asymptotical result of Ψᵣ (n, Cᵣ, L), and determined the asymptotically optimal result when ₁, , ₛ, r form an arithmetic progression and take certain values. We also study the generalized Turán 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.
Zhang et al. (Thu,) studied this question.