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Given a positive integer $n$ and an $r$-uniform hypergraph (or $r$-graph forshort) $F$, the Turan number $ex(n,F)$ of $F$ is the maximum number of edges inan $r$-graph on $n$ vertices that does not contain $F$ as a subgraph. Theextension $H^F $ of $F$ is obtained as follows: For each pair of vertices$v_i,v_j$ in $F$ not contained in an edge of $F$, we add a set $B_{ij}$ of$r-2$ new vertices and the edge $\{v_i,v_j\} \cup B_{ij}$, where the $B_{ij}$'s are pairwise disjoint over all such pairs $\{i,j\}$. Let $K^r_p$ denote thecomplete $r$-graph on $p$ vertices. For all sufficiently large $n$, wedetermine the Turan numbers of the extensions of a $3$-uniform $t$-matching, a$3$-uniform linear star of size $t$, and a $4$-uniform linear star of size $t$,respectively. We also show that the unique extremal hypergraphs are balancedblowups of $K^3_{3t-1}, K^3_{2t}$, and $K^4_{3t}$, respectively. Our resultsgeneralize the recent result of Hefetz and Keevash [7].

Turan numbers of extensions of some sparse hypergraphs via Lagrangians