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arXiv · 1406.6961

The typical structure of graphs with no large cliques

József Balogh·Neal Bushaw·Maurício Collares Neto·Hong Liu·Robert Morris·Maryam Sharifzadeh
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Abstract

In 1987, Kolaitis, Prömel 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 $r \leqslant (\log n)^{1/4}$. The proof combines a new (close to sharp) supersaturation version of the Erdős-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollobás and Simonovits.

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József Balogh, Neal Bushaw, Maurício Collares Neto, Hong Liu, Robert Morris, Maryam Sharifzadeh. 2015-04-30. The typical structure of graphs with no large cliques. https://arxiv.org/abs/1406.6961

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