Extremal results in sparse pseudorandom graphs
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Fox_Extremal results.pdf
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Author(s)
Conlon, David
Fox, Jacob
Zhao, Yufei
Date Issued
February 2014
Journal
Advances in Mathematics
Publisher
Elsevier
Citation
Conlon, David, Jacob Fox, and Yufei Zhao. “Extremal Results in Sparse Pseudorandom Graphs.” Advances in Mathematics 256 (May 2014): 206–290.
Version
Original manuscript
Abstract
Szemeredi's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rodl proved an analogue of Szemeredi's regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemeredi's regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a well-known open problem to prove a counting lemma in sparse graphs.
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 Rodl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several well-known combinatorial theorems, including the removal lemmas for graphs and groups, the Erdos–Stone–Simonovits theorem and Ramsey's theorem. These results extend and improve upon a substantial body of previous work.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1016/j.aim.2013.12.004
