Small subgraphs with large average degree

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• Oliver Janzer (Cambridge)
• Monday 10 October 2022, 14:30-15:30
• MR12.

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We study the fundamental problem of finding small dense subgraphs in a given graph. For a real number s>2, we prove that every graph on n vertices with average degree at least d contains a subgraph of average degree at least s on at most nd^{(log d)}{O_s(1)} vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. In addition, we show that every graph with n vertices and average degree at least n^{1-2/s+eps} contains a subgraph of average degree at least s on O_{eps,s}(1) vertices, which is also optimal up to the constant hidden in the O(.) notation, and resolves a conjecture of Verstraete. Joint work with Benny Sudakov and Istvan Tomon.

This talk is part of the Combinatorics Seminar series.

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