Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, June 03, 2014, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Hong Liu (University of Illinois at Urbana-Champaign)

The typical structure of graphs with no large cliques

In 1987, Kolaitis, Pr\"omel and Rothschild proved that, for every fixed $r \in \mathbb{N}$, almost every $n$-vertex $K_{r+1}$-free graph is $r$-partite. In this paper we extend this result to all functions $r = r(n)$ with $r \le (\log n)^{1/5}$. The proof combines a new (close to sharp) supersaturation version of the Erd\H{o}s-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollob\'as and Simonovits.

Joint work with J\'ozsef Balogh, Neal Bushaw, Mauricio Collares Neto, Robert Morris, Maryam Sharifzadeh

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