Szemerédi’s Lemma for the Analyst

Abstract

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.

Efficient rounding for the noncommutative grothendieck inequality

“…Theorem 3 is proved in Section 5.3 via an iterative application of Theorem 1, following the "energy decrement" strategy as formulated by Lovász and Szegedy [LS07] in the context of general weak regularity lemmas. Other than being a structural statement of interest in its own right, we show in Section 5.3 that Theorem 3 can be used to enhance the constant factor approximation of Theorem 1 to a PTAS for computing Opt…”

Geometric representations of graphs

“…The following was proved (in somewhat different form) in [140]. natural dimensionality, this dimension tends to give the right value.…”

Structural Properties of Sparse Graphs

“…It also let to manyfold applications and generalizations, see e.g. [66,65,72,113,41]. The closest to our topic covered in this paper is the recent development which is based on the study of homomorphisms of graphs (and structures).…”