The Brown-Erd\H{o}s-S\'os Conjecture for hypergraphs of large uniformity

TL;DR

This paper proves the Brown-Erdős-Sós Conjecture for hypergraphs with large uniformity, showing that dense linear hypergraphs of high uniformity contain small edge sets spanning few vertices.

Contribution

It establishes the conjecture for hypergraphs of large uniformity, extending previous results to a broader class of hypergraphs.

Findings

The conjecture holds for hypergraphs with sufficiently large uniformity.

Dense linear hypergraphs of large uniformity have small edge sets spanning few vertices.

The result depends on the hypergraph's density, uniformity, and size.

Abstract

We prove the well-known Brown-Erd\H{o}s-S\'os Conjecture for hypergraphs of large uniformity in the following form: any dense linear $r$-graph $G$ has $k$ edges spanning at most $(r-2)k+3$ vertices, provided the uniformity $r$ of $G$ is large enough given the linear density of $G$, and the number of vertices of $G$ is large enough given $r$ and $k$.