On generalized Turán problems for expansions

Given a graph $F$, the $r$-expansion $F^r$ of $F$ is the $r$-uniform hypergraph obtained from $F$ by inserting $r-2$ new distinct vertices in each edge of $F$. Given $r$-uniform hypergraphs $\mathcal{H}$ and $\mathcal{F}$, the generalized Turán number, denoted by $\textrm{ex}_r(n,\mathcal{H},\mathcal{F})$, is the maximum number of copies of $\mathcal{H}$ in an $n$-vertex $r$-uniform hypergraph that does not contain $\mathcal{F}$ as a subhypergraph. In the case where $r=2$ (i.e., the graph case), the study of generalized Turán problems was initiated by Alon and Shikhelman [\textit{J. Combin. Theory Series B.} 121 (2016) 146--172]. Motivated by their work, we systematically study generalized Turán problems for expansions and obtain several general and exact results. In particular, for the non-degenerate case, we determine the exact generalized Turán number for expansions of complete graphs, and establish the asymptotics of the generalized Turán number for expansions of the vertex-disjoint union of complete graphs. For the degenerate case, we establish the asymptotics of generalized Turán numbers for expansions of several classes of forests, including star forests, linear forests and star-path forests.