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Proof of a conjecture on induced subgraphs of Ramsey graphs
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abstract
An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in fact all Ramsey graphs must obey certain "richness" properties characteristic of random graphs. More than 25 years ago, Erd\H{o}s, Faudree and S\'{o}s conjectured that in any C-Ramsey graph there are $\Omega\left(n^{5/2}\right)$ 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.
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An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs
If a quadratic Bernoulli polynomial has a point probability much larger than 1/n, it is close to a quadratic form of low rank; a consequence bounds edge-count point probabilities in Ramsey graphs by n^{-1+o(1)}.
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