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<p>The celebrated Brown–Erdős–Sós conjecture states that for every fixed <i>e</i>, every 3-uniform hypergraph with Ω(<i>n</i><sup>2</sup>) edges contains <i>e</i> edges spanned by <i>e</i> + 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 <i>e</i> edges spanned by <i>e</i> + <i>O</i>(log <i>e</i>/ log log <i>e</i>) vertices.</p><p>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.</p>

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A new approach for the Brown–Erdős–Sós problem

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年2月3日

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  • Asaf Shapira,
  • Mykhaylo Tyomkyn

摘要

The celebrated Brown–Erdős–Sós conjecture states that for every fixed e, every 3-uniform hypergraph with Ω(n2) 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.