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Using Szemerédi’s Regularity Lemma to Prove Roth’s Theorem (2010)
| Venue: | UNIVERSITY OF GEORGIA REU- ADDITIVE COMBINATORICS- SUMMER 2010 |
BibTeX
@MISC{Bell10usingszemerédi’s,
author = {Stephanie Bell and Will Grodzicki},
title = {Using Szemerédi’s Regularity Lemma to Prove Roth’s Theorem},
year = {2010}
}
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Abstract
In this paper we use Szemerédi’s Regularity Lemma to show that we can take A, a subset of the integer lattice, of some density and show that it can be approximated by random graphs of the same density. Once we can think of these graphs as somewhat random, we can then prove that for N sufficiently large, one can always find a right isosceles triangle in A which can then be used to imply Roth’s Theorem. These ideas were borrowed from notes taken by Tom Sanders (March 14, 2004) on a course given by Tim Gowers titled Topics in Combinatorics. To begin, here are some definitions that will be used extensively in the following arguments. Definition 1 (Density). Let G be a graph and let A,B ⊂ V(G). The density of the pair (A,B), with edge set E(A,B) is defined to be d(A,B): = |E(A,B)| |A||B| Definition 2 (Epsilon Regular). Let ε> 0, then the pair (A,B) is said to be ε-regular if whenever A ′ ⊂ A with |A ′ | ≥ ε|A | and B ′ ⊂ B with |B ′ | ≥ ε|B | then |d(A ′,B ′ ) − d(A,B) | ≤ ε Definition 3 (Epsilon Regular Partitions). For a graph G with N vertices we shall say that a partition χ: = {X1,...,Xj} of V (G) is ε-regular if (i,j):(Xi,Xj) is not ε−regular
