Generalized Turán problems for Berge hypergraphs

Let $\mathcal{H}$ be a hypergraph and $F$ be a graph. If there exists a bijection between the hyperedges of $\mathcal{H}$ and the edges of $F$ such that each hyperedge contains its image, then we say that $\mathcal{H}$ is a $\mathit{\text{Berge}}\mathit{\text{copy}}$ of $F$, and the collection of Berge copies of $F$ is denoted by Berge-$F$. Given $r$-graphs $\mathcal{F}$ and $\mathcal{H}$, the generalized hyper-Turán number ${\text{ex}}_r(n,\mathcal{H},\mathcal{F})$ is the maximum number of copies of $\mathcal{H}$ in $n$-vertex $\mathcal{F}$-free $r$-graphs. We study ${\text{ex}}_r(n,\mathcal{H},\text{Berge-}F)$. For general $\mathcal{H}$, we connect this problem to counting copies of the shadow graph of $\mathcal{H}$ in $F$-free graphs and obtain several exact results. In particular, we show that for any hypergraph $\mathcal{H}$, if $k$ is sufficiently large, then ${\text{ex}}_r(n,\mathcal{H},\text{Berge-}{K}_k)$ is achieved by the balanced complete $(k-1)$-partite $r$-graph, generalizing a result of Morrison, Nir, Norin, Rza{ż}ewski and Wesolek [$\mathit{\text{Journal}}\mathit{\text{of}}\mathit{\text{Combinatorial}}\mathit{\text{Theory,}}\mathit{\text{Series}}\mathit{\text{B}}$, 162 (2023) 231–243] to the case of hypergraphs. We show that ${\text{ex}}_r(n,K_s^r,\text{Berge-}F)\le {\text{ex}}_s(n,\text{Berge-}F)$ and present sufficient conditions for equality. We also consider the connected generalized Turán number for Berge paths.