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=\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)=Θr,L(nˢ) if and only if ₁,,ₛ,r form an arithmetic progression. In this paper, we show that Ψᵣ(n,F,L)=or,L(nˢ) 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.