Abstract: Given an $r$-uniform hypergraph $\mathcal{F}$, the {\em Tur\'{a}n number} of $\mathcal{F}$, denoted by $\mathrm{ex}_{r}(n,\mathcal{F})$, is the maximum number of edges in an $\mathcal{F}$-free $r$-uniform hypergraph on $n$ vertices. For $r\ge 3$, determining $\mathrm{ex}_{r}(n,\mathcal{F})$ is known to be notoriously hard especially when $\mathrm{ex}_{r}(n,\mathcal{F})=o(n^r)$. For a graph $F$, the expansion of $F$, denoted by $F^{+}$, is an $r$-uniform hypergraph by adding $r-2$ new elements to each edge of $F$; and the Berge copy of $F$, denoted by Berge-$F$, is an $r$-uniform hypergraph $\mathcal{H}$ with $V(F) \subseteq V(\mathcal{H})$ satisfying that there is a bijection $f$ from $E(F)$ to $E(\mathcal{H})$ such that $e\subseteq f(e)$ for every $e \in E(F)$. In this paper, we determine the Tur\'{a}n numbers of the expansion, and the family of all Berge copy of disjoint union of stars. Both generalize results given by Khormali and Palmer [European J. of Combin. 102 (2022)].

Key words: Turán , number, Star, Expansion, Berge cop