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In this expository article, we give a detailed proof of a qualitative version of the Mallaris-Shelah regularity lemma for stable graphs (Malliaris and Shelah in Trans. Am. Math. Soc. 366(3):1551–1585, 2014) using only basic local stability theory and an ultraproduct construction. This proof strategy was first established by Malliaris and Pillay (Proc. Am. Math. Soc. 144(4):1761–1765, 2016), and later simplified by Pillay (Bull. Symb. Log. 26(2):103–117, 2020). We provide some further simplifications, and also explain how the pseudofinite approach can be used to obtain a qualitative strengthening (in comparison to Malliaris and Pillay (Proc. Am. Math. Soc. 144(4):1761–1765, 2016) and Pillay (Bull. Symb. Log. 26(2):103–117, 2020) in terms of “functional error”. To illustrate the extra leverage obtained by functional error, we give an elementary argument for extracting equipartitions from arbitrary partitions.

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Pseudofinite Proofs of the Stable Graph Regularity Lemma

  • 章节

  • 发表日期：

    2026年

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  • G. Conant,
  • C. Terry

摘要

In this expository article, we give a detailed proof of a qualitative version of the Mallaris-Shelah regularity lemma for stable graphs (Malliaris and Shelah in Trans. Am. Math. Soc. 366(3):1551–1585, 2014) using only basic local stability theory and an ultraproduct construction. This proof strategy was first established by Malliaris and Pillay (Proc. Am. Math. Soc. 144(4):1761–1765, 2016), and later simplified by Pillay (Bull. Symb. Log. 26(2):103–117, 2020). We provide some further simplifications, and also explain how the pseudofinite approach can be used to obtain a qualitative strengthening (in comparison to Malliaris and Pillay (Proc. Am. Math. Soc. 144(4):1761–1765, 2016) and Pillay (Bull. Symb. Log. 26(2):103–117, 2020) in terms of “functional error”. To illustrate the extra leverage obtained by functional error, we give an elementary argument for extracting equipartitions from arbitrary partitions.