Induced Ramsey-type theorems
Jacob Fox,Benny Sudakov +1 more
TL;DR: In this paper, a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph is presented, which can be used to extend and improve the earlier results of Rodl, Erdős-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham, and Łuczak-Rucinski.
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About: This article is published in Advances in Mathematics. The article was published on 20 Dec 2008. and is currently open access. The article focuses on the topics: Ramsey theory & Ramsey's theorem.
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Citations
•Journal Article
Simulating Independence: New Constructions of Condensers, Ramsey Graphs, Dispersers, and Extractors.
Abstract: We present new explicit constructions of deterministic randomness extractors, dispersers and related objects. We say that a distribution X on binary strings of length n is a δ-source if X assigns probability at most 2−δn to any string of length n. For every δ>0, we construct the following poly(n)-time computable functions:2-source disperser: D:({0, 1}n)2 → {0, 1} such that for any two independent δ-sources X1,X2 we have that the support of D(X1,X2) is {0, 1}.Bipartite Ramsey graph: Let N=2n. A corollary is that the function D is a 2-coloring of the edges of KN,N (the complete bipartite graph over two sets of N vertices) such that any induced subgraph of size Nδ by Nδ is not monochromatic.3-source extractor:E:({0, 1}n)3→ {0, 1} such that for any three independent δ-sources X1,X2,X3 we have that E(X1,X2,X3) is o(1)-close to being an unbiased random bit.No previous explicit construction was known for either of these for any δ
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•Posted Content
Recent developments in graph Ramsey theory
TL;DR: The existence of the Ramsey number has been known since 1930 but their quantitative behaviour is still not well understood as mentioned in this paper. But there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by many advances across extremal combinatorics.
137
Recent developments in graph Ramsey theory.
David Conlon,Jacob Fox,Benny Sudakov +2 more
- 01 Jan 2015
TL;DR: There has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics.
Extremal results in sparse pseudorandom graphs
TL;DR: A new counting lemma is proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and Rodl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph and to prove sparse extensions of several well-known combinatorial theorems.
86
On two problems in graph Ramsey theory
TL;DR: This work improves the upper bound on the existence of a constant c such that, for any graph H on n vertices, rind(H) ≤ 2cnlogn, and moves a step closer to proving this conjecture.
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