Extremal results in sparse pseudorandom graphs
Szemerédi'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 Rödl proved an analogue of Szemerédi'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 Szemerédi'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 Rödl, 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 Erdős–Stone–Simonovits theorem and Ramsey's theorem. These results extend and improve upon a substantial body of previous work.
Additional Information© 2014 Elsevier. Under an Elsevier user license. Received 4 June 2012; accepted 14 December 2013; available online 25 February 2014. Conlon research supported by a Royal Society University Research Fellowship. Fox research supported by a Simons Fellowship, a Packard Fellowship, an Alfred P. Sloan Fellowship, an MIT NEC Corporation Award, and NSF grant DMS-1069197. Zhao research supported by an Akamai Presidential Fellowship and a Microsoft Research PhD Fellowship. We thank the referees for a number of helpful comments which improved the manuscript.
Submitted - 1204.6645.pdf