Large cliques and independent sets all over the place
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Date
2021-08
Publication Type
Journal Article
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yes
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Abstract
We study the following question raised by Erdos and Hajnal in the early 90's. Over all n-vertex graphs G what is the smallest possible value of m for which any m vertices of G contain both a clique and an independent set of size log n? We construct examples showing that m is at most 2(2(log log n)1/2+o(1)) obtaining a twofold sub-polynomial improvement over the upper bound of about root n coming from the natural guess, the random graph. Our (probabilistic) construction gives rise to new examples of Ramsey graphs, which while having no very large homogenous subsets contain both cliques and independent sets of size log n in any small subset of vertices. This is very far from being true in random graphs. Our proofs are based on an interplay between taking lexicographic products and using randomness.
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published
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Journal / series
Volume
149 (8)
Pages / Article No.
3145 - 3157
Publisher
American Mathematical Society
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Organisational unit
03993 - Sudakov, Benjamin / Sudakov, Benjamin
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Funding
196965 - Problems in Extremal and Probabilistic Combinatorics (SNF)