Turan numbers of extensions of some sparse hypergraphs via Lagrangians

TL;DR

This paper determines the Turan numbers for extensions of certain sparse hypergraphs using Lagrangian methods, identifying extremal structures for large n and generalizing recent results.

Contribution

It provides exact Turan numbers and extremal hypergraphs for extensions of specific sparse hypergraphs, expanding previous work with new classes and methods.

Findings

Exact Turan numbers for extensions of 3-uniform t-matchings

Exact Turan numbers for extensions of 3-uniform linear stars

Exact Turan numbers for extensions of 4-uniform linear stars

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 $K^r_p$ 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 $K^3_{3t-1}, K^3_{2t}$, and $K^4_{3t}$, respectively. Our results generalize the recent result of Hefetz and Keevash [7].