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Continuous stable regularity

Nicolas Chavarria, Gabriel Conant, Anand Pillay

Abstract

We prove an analytic version of the stable graph regularity lemma from \cite{MaSh}, which applies to stable functions $f\colon V\times W\to [0,1]$. Our methods involve continuous model theory and, in particular, results on the structure of local Keisler measures for stable continuous formulas. Along the way, we develop some basic tools around ultraproducts of metric structures and linear functionals on continuous formulas, and we also describe several concrete families of examples of stable functions.

Continuous stable regularity

Abstract

We prove an analytic version of the stable graph regularity lemma from \cite{MaSh}, which applies to stable functions . Our methods involve continuous model theory and, in particular, results on the structure of local Keisler measures for stable continuous formulas. Along the way, we develop some basic tools around ultraproducts of metric structures and linear functionals on continuous formulas, and we also describe several concrete families of examples of stable functions.

Paper Structure

This paper contains 15 sections, 30 theorems, 48 equations.

Table of Contents

  1. Stable functions
  2. Preliminaries on continuous logic
  3. Basic notions
  4. Linear functionals
  5. Linear functionals on metric structures
  6. Keisler measures on stable formulas
  7. Model-theoretic structure of stable formulas
  8. Stable regularity for functions on finite sets
  9. Approximating the counting measure
  10. Proof of Theorem \ref{['thm:main-finite-thm']}
  11. Further results
  12. Equipartitions
  13. The case of graphs
  14. Function decomposition
  15. Bipartite versus symmetric

Key Result

Theorem A

Let $V$ and $W$ be finite sets, and suppose $f\colon V\times W\to [0,1]$ is a $(k,\delta)$-stable function. Then for any $\varepsilon>0$ and any "decay" function $\sigma\colon \mathbb{N}\to (0,1)$, there are partitions $V=V_0\cup V_1\cup\ldots\cup V_m$ and $W=W_0\cup W_1\cup\ldots\cup W_n$, with $m,

Theorems & Definitions (88)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • proof
  • Corollary 1.4
  • proof
  • Remark 1.5
  • ...and 78 more