A new approach for the Brown–Erdős–Sós problem

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Abstract 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.