Turan numbers of extensions of some sparse hypergraphs via Lagrangians

Abstract

Given a positive integer $n$ and an $r$-uniform hypergraph (or $r$-graph for short) $F$, the Turan number $ex(n,F)$ of $F$ is the maximum number of edges in an $r$-graph on $n$ vertices that does not contain $F$ as a subgraph. The extension $H^F$ 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 $K_p^r$ denote the complete $r$-graph on $p$ vertices. For all sufficiently large $n$, we determine 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 balanced blowups of $K_{3t-1}^3,K_{2t}^3$, and $K_{3t}^4$, respectively. Our results generalize the recent result of Hefetz and Keevash [7].

21 pages

Cited by in corpus (1)

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