On a conjecture of Hefetz and Keevash on Lagrangians of intersecting hypergraphs and Turán numbers

Abstract

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 $p\ge |V(F)|$, 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 Turán number of the extension of $M_2^r$ is $\frac{1}{r}n\cdot \left(\genfrac{}{}{0ex}{}{\frac{r-1}{r}n}{r-1}\right)$ 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 $\frac{1}{r}n\cdot \left(\genfrac{}{}{0ex}{}{\frac{r-1}{r}n}{r-1}\right)$ edges, then $G$ is isomorphic to $S^r(n)$. In this paper, we confirm the above conjecture for $r=4$.

25 pages. arXiv admin note: text overlap with arXiv:1307.8423 by other authors

References in corpus (1)

• Turan numbers of extensions of some sparse hypergraphs via Lagrangians

Cited by in corpus (1)

• Lagrangian densities of short 3-uniform linear paths and Turán numbers of their extensions