Generalized Tur\'an 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 \textit{Berge 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-Tur\'{a}n 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{\.z}ewski and Wesolek [\textit{Journal of Combinatorial Theory, Series 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 Tur\'{a}n number for Berge paths.