A new approach for the Brown-Erdos-Sos problem

TL;DR

This paper introduces a novel approach to the Brown-Erdős-Sós conjecture, reducing it to a related extremal graph theory problem, potentially enabling a near-complete resolution.

Contribution

It presents a new method that shifts the problem to a different conjecture, bypassing complex regularity techniques used in prior work.

Findings

Reduces the conjecture to a variant of a known extremal graph theory conjecture.

Achieves a near-complete resolution up to an absolute additive constant.

Provides a different perspective that may simplify future proofs.

Abstract

The celebrated Brown-Erd\H{o}s-S\'os conjecture states that for every fixed $e$, every $3$-uniform hypergraph with $\Omega(n^2)$ edges contains $e$ edges spanned by $e+3$ vertices. Up to this date all the approaches towards resolving this problem relied on highly involved applications of the hypergraph regularity method, and yet they supplied only approximate versions of the conjecture, producing $e$ edges spanned by $e+O(\log e/\log \log e)$ vertices. In this short paper we describe a completely different approach, which reduces the problem to a variant of another well-known conjecture in extremal graph theory. A resolution of the latter would resolve the Brown-Erd\H{o}s-S\'os conjecture up to an absolute additive constant.