Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, October 18, 2018, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Tuan Tran

Subgraph statistics

Consider integers k and l such that 0 < l < k(k-1)/2. Given a large graph G, what is the fraction of k-vertex subsets of G which span exactly l edges? When G is empty or complete, and l is zero or k(k-1)/2, this fraction can be exactly 1. On the other hand if l is not one of these extreme values, then by Ramsey's theorem, this fraction is strictly smaller than 1. The systematic study of the above question was recently initiated by Alon, Hefetz, Krivelevich and Tyomkyn who proposed several natural conjectures. In this talk we present solutions of two of them (one assymptotically) and make first steps towards analogous questions for hypergraphs. Our proofs involve some Ramsey-type arguments, and a number of different probabilistic tools, such as polynomial anticoncentration inequalities, hypercontractivity, and a coupling argument. Joint work with M. Kwan and B. Sudakov.

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