A Tur\'an theorem for extensions via an Erd\H{o}s-Ko-Rado theorem for Lagrangians

TL;DR

This paper determines the Turán number for extensions of specific hypergraphs using an Erdős-Ko-Rado type theorem for Lagrangians, settling a conjecture and advancing hypergraph extremal theory.

Contribution

It establishes the Turán number for extensions of two disjoint edges and proves that intersecting r-graphs maximize Lagrangians when principally intersecting for r ≥ 4.

Findings

Turán number for extensions of two disjoint edges determined

Lagrangian of intersecting r-graphs maximized by principally intersecting graphs for r ≥ 4

Settles a conjecture of Hefetz and Keevash

Abstract

The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $(r-2)$ new vertices. In this paper we determine the Tur\'an number of the extension of an $r$-graph consisting of two vertex-disjoint edges, settling a conjecture of Hefetz and Keevash, who previously determined this Tur\'an number for $r=3$. As the key ingredient of the proof we show that the Lagrangian of intersecting $r$-graphs is maximized by principally intersecting $r$-graphs for $r \geq 4$.