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Status: Unchecked ×Keyword: pseudorandom graphs ×Clear all

4 claims from 2 papers

1. Mathematics › Limits and Structures in Graph Theory

   Extremal results in sparse pseudorandom graphs

   Conlon, Fox and Zhao · Advances in Mathematics · 2014

   The paper proves a counting lemma for sparse graphs that complements Kohayakawa and Rödl's sparse regularity lemma, and uses it to extend several well-known combinatorial theorems to sparse settings.

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   1. UncheckedThe paper's main advance is a new counting lemma that lets one count small graphs inside regular subgraphs of a sufficiently pseudorandom graph.“The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and Rödl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph.”
   2. UncheckedThe authors prove versions of several classic combinatorial theorems that also apply to sparse graphs, using a new counting lemma.“We use this to prove sparse extensions of several well-known combinatorial theorems, including the removal lemmas for graphs and groups, the Erdős-Stone-Simonovits theorem and Ramsey's theorem.”

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.”