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 $p\geq |V(F)|$, we define the {\em extension} of $F$, denoted by $H_{p}^{F}$ as follows: Label the vertices of $F$ as $v_1,\dots,v_{|V(F)|}$. Add new vertices $v_{|V(F)|+1},\dots,v_{p}$. 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 $\{v_i,v_j\} \cup B_{ij}$, where the $B_{ij}$ 's are pairwise disjoint over all such pairs $\{i,j\}$. Hefetz and Keevash conjectured that the Tur\'an number of the extension of $M_2^r$ is ${1 \over r}n\cdot{{r-1\over r}n \choose r-1}$ for $r \ge 4$ and sufficiently large $n$. Moreover, if $n$ is sufficiently large and $G$ is an $H_{2r}^{M_2^r}$-free $r$-graph with $n$ vertices and ${1 \over r}n\cdot{{r-1\over r}n \choose r-1}$ edges, then $G$ is isomorphic to $S^r(n)$. In this paper, we confirm the above conjecture for $r=4$.