Proof of a conjecture on induced subgraphs of Ramsey graphs

Author:

Kwan Matthew,Sudakov Benny

Abstract

An n n -vertex graph is called C C -Ramsey if it has no clique or independent set of size C log n C\log n . All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research toward showing that in fact all Ramsey graphs must obey certain “richness” properties characteristic of random graphs. More than 25 years ago, Erdős, Faudree, and Sós conjectured that in any C C -Ramsey graph there are Ω ( n 5 / 2 ) \Omega (n^{5/2}) induced subgraphs, no pair of which have the same numbers of vertices and edges. Improving on earlier results of Alon, Balogh, Kostochka, and Samotij, in this paper we prove this conjecture.

Funder

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

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