Large cliques and independent sets all over the place

Noga Alon, Matija Bucić, Benny Sudakov

Research output: Contribution to journalArticlepeer-review

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 22(log log n)1/2+o(1) obtaining a twofold sub-polynomial improvement over the upper bound of about √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.

Original languageEnglish
Pages (from-to)3145-3157
Number of pages13
JournalProceedings of the American Mathematical Society
Volume149
Issue number8
DOIs
StatePublished - 2021

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