Tur\' an number for bushes

Abstract: Let $ a,b \in {\bf Z}^+$, $r=a + b$, and let $T$ be a tree with parts $U = {u_1,u_2,\dots,u_s}$ and $V = {v_1,v_2,\dots,v_t}$. Let $U_1, \dots ,U_s$ and $V_1, \dots, V_t$ be disjoint sets, such that {$|U_i|=a$ and $|V_j|=b$ for all $i,j$}. The {\em $(a,b)$-blowup} of $T$ is the $r$-uniform hypergraph with edge set $ {{U_i \cup V_j : u_iv_j \in E(T)}.}$ We use the $\Delta$-systems method to prove the following Tur\' an-type result. Suppose $a,b,s \in {\bf Z}^+$, $r=a+b\geq 3$,{ $a\geq 2$,} and $T$ is a fixed tree of diameter $4$ in which the degree of the center vertex is $s $. Then there exists a $C=C(r,s ,T)>0$ such that $ |\mathcal{H}|\leq (s -1){n\choose r-1} +Cn^{r-2}$ for every $n$-vertex $r$-uniform hypergraph $\mathcal{H}$ {not containing an $(a,b)$-blowup of $T$}. This is {asymptotically exact} when $s \leq |V(T)|/2$. A stability result is also presented.