Szemerédi’s Lemma for the Analyst
نویسندگان
چکیده
Szemerédi’s Regularity Lemma is a fundamental tool in graph theory: it has many applications to extremal graph theory, graph property testing, combinatorial number theory, etc. The goal of this paper is to point out that Szemerédi’s Lemma can be thought of as a result in analysis. We show three different analytic interpretations.
منابع مشابه
Szemerédi’s Proof of Szemerédi’s Theorem
In 1975, Szemerédi famously established that any set of integers of positive upper density contained arbitrarily long arithmetic progressions. The proof was extremely intricate but elementary, with the main tools needed being the van der Waerden theorem and a lemma now known as the Szemerédi regularity lemma, together with a delicate analysis (based ultimately on double counting arguments) of l...
متن کاملSzemerédi’s Regularity Lemma for Sparse Graphs
A remarkable lemma of Szemerédi asserts that, very roughly speaking, any dense graph can be decomposed into a bounded number of pseudorandom bipartite graphs. This far-reaching result has proved to play a central rôle in many areas of combinatorics, both ‘pure’ and ‘algorithmic.’ The quest for an equally powerful variant of this lemma for sparse graphs has not yet been successful, but some prog...
متن کاملInformation Theoretic Interpretation of Szemerédi’s Regularity Lemma
Szemerédi’s regularity lemma is a fundamental tool in the theory of very large and dense graphs. In particular, it can be viewed as a structure theorem for arbitrary dense graphs which decomposes such graphs into a large number of parts such that the sub-graphs between these parts are random-like (simple-structured) on each pair, except for a small number of pairs. The regularity lemma has seve...
متن کاملSzemerédi’s Regularity Lemma and Quasi-randomness
The first half of this paper is mainly expository, and aims at introducing the regularity lemma of Szemerédi. Among others, we discuss an early application of the regularity lemma that relates the notions of universality and uniform distribution of edges, a form of ‘pseudorandomness’ or ‘quasi-randomness’. We then state two closely related variants of the regularity lemma for sparse graphs and ...
متن کاملSzemerédi’s Regularity Lemma
Szemerédi’s Regularity Lemma is an important result in extremal graph theory. Roughly speaking, the lemma states that every graph can be approximated by random graphs; that is, the vertex set of every graph can be split into equal size subsets such that the distribution of the edges between almost any two of these subsets is pseudorandom. The Regularity Lemma has already proved to be a powerful...
متن کامل