Tur\'an numbers of extensions

TL;DR

This paper determines the maximum number of edges in certain hypergraphs called extensions, building on previous work on Turán densities, using stability methods and new analytical tools.

Contribution

It provides exact Turán numbers for specific hypergraph extension families, advancing the understanding of extremal hypergraph configurations.

Findings

Calculated Turán numbers for two families of hypergraph extensions.

Applied classical stability techniques and introduced new tools.

Extended previous results on Turán densities to exact numbers.

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. Keevash and Sidorenko~ have previously determined Tur\'an densities of two families of hypergraph extensions. We determine the Tur\'an numbers for these families, using classical stability techniques and new tools introduced in our earlier paper.