Small Cores in 3-uniform Hypergraphs

TL;DR

This paper proves that dense large 3-uniform hypergraphs necessarily contain small cores of at most 15 vertices, advancing understanding of their structural properties and connecting to longstanding conjectures.

Contribution

It establishes a new bound on the size of cores in dense 3-uniform hypergraphs, improving previous results and linking to open conjectures.

Findings

Dense hypergraphs contain small cores of at most 15 vertices.

The result is close to optimal, as improving it would resolve a long-standing conjecture.

The paper connects core existence to extremal properties of hypergraphs.

Abstract

The main result of this paper is that for any $c>0$ and for large enough $n$ if the number of edges in a 3-uniform hypergraph is at least $cn^2$ then there is a core (subgraph with minimum degree at least 2) on at most 15 vertices. We conjecture that our result is not sharp and 15 can be replaced by 9. Such an improvement seems to be out of reach, since it would imply the following case of a long-standing conjecture by Brown, Erd\H os, and S\'os; if there is no set of 9 vertices that span at least 6 edges of a 3-uniform hypergraph then it is sparse.