On Erdos' extremal problem on matchings in hypergraphs

TL;DR

This paper proves Erdős's conjecture for 3-uniform hypergraphs with large enough n, showing the maximum number of edges occurs in specific extremal hypergraphs when the largest matching has s edges.

Contribution

The paper confirms Erdős's conjecture for 3-uniform hypergraphs for sufficiently large n, identifying the extremal structures that maximize edges.

Findings

Erdős's conjecture holds for 3-uniform hypergraphs with large n.

Maximum edges are achieved by specific extremal hypergraphs.

The result advances understanding of extremal hypergraph configurations.

Abstract

In 1965 Erd\H{o}s conjectured that the number of edges in k-uniform hypergraphs on n vertices in which the largest matching has s edges is maximized for hypergraphs of one of two special types. We settled this conjecture in the affirmative for k=3 and n is large enough.