Regularity lemmas in a Banach space setting

TL;DR

This paper extends the concept of Szemerédi's regularity lemma to Banach spaces, providing new compactness results that generalize previous Hilbert space interpretations and apply to a broader functional analysis context.

Contribution

It introduces several compactness results in Banach spaces, generalizing earlier Hilbert space-based regularity lemmas and graph limit theories.

Findings

Proves new compactness results in Banach spaces.

Generalizes Lovász and Szegedy's Hilbert space interpretation.

Extends graph limit theory to Banach space setting.

Abstract

Szemer\'edi's regularity lemma is a fundamental tool in extremal graph theory, theoretical computer science and combinatorial number theory. Lov\'asz and Szegedy [L. Lov\'asz and B. Szegedy: Szemer\'edi's Lemma for the analyst, Geometric and Functional Analysis 17 (2007), 252-270] gave a Hilbert space interpretation of the lemma and an interpretation in terms of compact- ness of the space of graph limits. In this paper we prove several compactness results in a Banach space setting, generalising results of Lov\'asz and Szegedy as well as a result of Borgs, Chayes, Cohn and Zhao [C. Borgs, J.T. Chayes, H. Cohn and Y. Zhao: An Lp theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions, arXiv preprint arXiv:1401.2906 (2014)].