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Keyword: Szemerédi regularity lemma ×Clear all

4 claims from 2 papers

1. Mathematics › Limits and Structures in Graph Theory

   Hypergraph regularity and the multidimensional Szemerédi theorem

   Gowers · Annals of Mathematics · 2007

   The paper proves hypergraph versions of the regularity and counting lemmas, and uses them to give a combinatorial proof of the multidimensional Szemerédi theorem with an explicit bound.

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   1. UncheckedThe paper proves versions for hypergraphs of Szemerédi's regularity lemma and its counting lemma, both of which were previously known for ordinary graphs.“We prove analogues for hypergraphs of Szemerédi's regularity lemma and the associated counting lemma for graphs.”
   2. UncheckedGowers says hypergraph regularity gives the first combinatorial proof of the multidimensional Szemerédi theorem, and the first with an explicit bound.“As an application, we give the first combinatorial proof of the multidimensional Szemerédi theorem of Furstenberg and Katznelson, and the first proof that provides an explicit bound.”

2. Mathematics › Limits and Structures in Graph Theory

   Induced Ramsey-type theorems

   Fox and Sudakov · arXiv (Cornell University) · 2007

   The paper gives a unified approach to Ramsey-type theorems for graphs with a forbidden induced subgraph, extending earlier results and also showing that pseudo-random graphs have strong induced Ramsey properties.

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   1. UncheckedThe paper's proofs rest on a simple lemma that can stand in for Szemerédi's regularity lemma and, as the authors state, gives much better bounds.“The proofs are based on a simple lemma (generalizing one by Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's regularity lemma, thereby giving much better bounds.”
   2. UncheckedThe paper's approach can also be used to show that pseudo-random graphs have strong induced Ramsey properties.“The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties.”