Regularity and removal lemmas and their applications

In this thesis, we analyze the regularity method pioneered by Szemerédi, and also discuss one of its prevalent applications, the removal lemma. First, we prove a new lower bound on the number of parts required in a version of Szemerédi's regularity lemma, determining the order of the tower height in that version up to a constant factor. This addresses a question of Gowers. Next, we turn to algorithms. We give a fast algorithmic Frieze-Kannan (weak) regularity lemma that improves on previous running times. We use this to give a substantially faster deterministic approximation algorithm for counting subgraphs. Previously, only an exponential dependence of the running time on the error parameter was known; we improve it to a polynomial dependence. We also revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for some co-NP-complete problems. We show how to use the Frieze-Kannan regularity lemma to approximate the regularity of a pair of vertex sets. We also show how to quickly find, for each [epsilon]' > [epsilon], an [epsilon]'-regular partition with k parts if there exists an [epsilon]-regular partition with k parts. After studying algorithms, we turn to the arithmetic setting. Green proved an arithmetic regularity lemma, and used it to prove an arithmetic removal lemma. The bounds obtained, however, were tower-type, and Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. The previous best known bound was tower-type with a logarithmic tower height. We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in Fn/p. Finally, we give a new proof of a regularity lemma for permutations, improving the previous tower-type bound on the number of parts to an exponential bound.

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2017

Author dc:creator

• Lovász, László Miklós

Advisor dc:contributor.advisor

• Jacob Fox.

• Mathematics.

Statement dc:rights

• MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/112899

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/112899