A new approach for the Brown–Erdős–Sós problem

The celebrated Brown–Erdős–Sós conjecture states that for every fixed e , every 3-uniform hypergraph with Ω( 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–Erdős–Sós conjecture up to an absolute additive constant.