The Regularity Lemma and Graph Theory

Szemerédi's regularity lemma revisited

Szemerédi’s regularity lemma is a basic tool in graph theory, and also plays an important role in additive combinatorics, most notably in proving Szemerédi’s theorem on arithmetic progressions [19], [18]. In this note we revisit this lemma from the perspective of probability theory and information theory instead of graph theory, and observe a slightly stronger variant of this lemma, related to ...

Szemerédi’s Lemma for the Analyst

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.

Note on the 3-graph counting lemma

Szemerédi’s regularity lemma proved to be a powerful tool in extremal graph theory. Many of its applications are based on the so-called counting lemma: if G is a kpartite graph with k-partition V1∪ · · ·∪Vk, |V1| = · · · = |Vk| = n, where all induced bipartite graphs G[Vi, Vj ] are (d, ε)-regular, then the number of k-cliques Kk in G is d( k 2)nk(1± o(1)). Frankl and Rödl extended Szemerédi’s r...

Szemerédi's Regularity Lemma and Its Applications in Graph Theory

A Szemerédi-type regularity lemma in abelian groups

Szemerédi’s regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemerédi’s regularity lemma in the context of abelian groups and use it to derive some results in additive number theory. The simplest is a structure theorm for sets which are almost sum-free. If A ⊆ {1, . . . , N} has δN2 triples (a1, a2, a3)...