Abstract
Szemerédi's regularity lemma is a fundamental tool in extremal graph theory, theoretical computer science and combinatorial number theory. Lovász and Szegedy [Lovász, L., and B. Szegedy, Szemerédi's Lemma for the analyst, Geom. Funct. Anal. 17 (2007), 252–270] gave a Hilbert space interpretation of the lemma and an interpretation in terms of compactness of the space of graph limits. In this paper we prove several compactness results in a Banach space setting, generalising results of Lovász and Szegedy [Lovász, L., and B. Szegedy, Szemerédi's Lemma for the analyst, Geom. Funct. Anal. 17 (2007), 252–270] as well as a result of Borgs, Chayes, Cohn and Zhao [Borgs, C., 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)].
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