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

Date and Time: Monday, June 26, 2017, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Asaf Ferber (MIT)

Counting H-free graphs

A central problem in extremal graph theory is the problem of counting the number of H-free graphs on n vertices. While for non-bipartite graphs H this problem is essentially solved, for the bipartite case only very little is known. One main difficulty is that the extremal function , ex(n,H), for a bipartite H is generally unknown. Motivated by a recent result of Balogh, Liu and Sharifzadeh, we show that if ex(n,H)=O(n^{\alpha}) for some fixed \alpha, then the number of H-free graphs is at most 2^{O(n^{\alpha})}. In particular, if ex(n,H)=\Theta(n^{\alpha}) (like in all known examples), then our result is tight up to the constant in the exponent. Our result reproves all the previous known results on this problem and in addition solves a problem of Mubayi and Wang which has been recently solved independently by Balogh, Narayanan and Skokan. This is a joint work with Gweneth Anne McKinley and Wojtek Samotij.

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