On a conjecture of Hefetz and Keevash on Lagrangians of intersecting hypergraphs and Tur\'an numbers

Let $S^r(n)$ be the $r$-graph on $n$ vertices with parts $A$ and $B$, where the edges consist of all $r$-tuples with $1$ vertex in $A$ and $r-1$ vertices in $B$, and the sizes of $A$ and $B$ are chosen to maximise the number of edges. Let $M_t^r$ be the $r$-graph with $t$ pairwise disjoint edges. Given an $r$-graph $F$ and a positive integer $p\ge \mathrm{\mid }V(F)\mathrm{\mid }$, we define the {\em extension} of $F$, denoted by $H_p^F$ as follows: Label the vertices of $F$ as $v_1,\dots ,v_{\mathrm{\mid }V(F)\mathrm{\mid }}$. Add new vertices $v_{\mathrm{\mid }V(F)\mathrm{\mid }+1},\dots ,v_p$. For each pair of vertices $v_i,v_j,1\le i<j\le p$ not contained in an edge of $F$, we add a set $B_{ij}$ of $r-2$ new vertices and the edge ${v_i,v_j} \cup B_{ij}$,wherethe$B_{ij}${}^{\mathrm{\prime }}sarepairwisedisjointoverallsuchpairs${i,j}$. Hefetz and Keevash conjectured that the Tur'an number of the extension of $M_2^r$ is $\frac{1}{r}n\cdot \left(\genfrac{}{}{0px}{}{\frac{r-1}{r}n}{r-1}\right)$ for $r \ge 4$andsufficientlylarge$n$\mathrm{.}Moreover,if$n$issufficientlylargeand$G$isan$H_{2r}^{M_2^r}$-free$r$-graphwith$n$verticesand${1 \over r}n\cdot{{r-1\over r}n \choose r-1}$edges,then$G$isisomorphicto$S^r(n)$. In this paper, we confirm the above conjecture for $r=4$.