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Intersecting sets in probability spaces and Shelah's classification

Artem Chernikov, Henry Towsner

Abstract

For $n \in \mathbb{N}$ and $\varepsilon > 0$, given a sufficiently long sequence of events in a probability space all of measure at least $\varepsilon$, some $n$ of them will have a common intersection. A more subtle pattern: for any $0 < p < q < 1$, we cannot find events $A_i$ and $B_i$ so that $μ\left( A_i \cap B_j \right) \leq p$ and $μ\left( A_j \cap B_i\right) \geq q$ for all $1 < i < j < n$, assuming $n$ is sufficiently large. This is closely connected to model-theoretic stability of probability algebras. We survey some results from our recent work on more complicated patterns that arise when our events are indexed by multiple indices. In particular, how such results are connected to higher arity generalizations of de Finetti's theorem in probability, structural Ramsey theory, hypergraph regularity in combinatorics, and model theory.

Intersecting sets in probability spaces and Shelah's classification

Abstract

For and , given a sufficiently long sequence of events in a probability space all of measure at least , some of them will have a common intersection. A more subtle pattern: for any , we cannot find events and so that and for all , assuming is sufficiently large. This is closely connected to model-theoretic stability of probability algebras. We survey some results from our recent work on more complicated patterns that arise when our events are indexed by multiple indices. In particular, how such results are connected to higher arity generalizations of de Finetti's theorem in probability, structural Ramsey theory, hypergraph regularity in combinatorics, and model theory.

Paper Structure

This paper contains 13 sections, 1 theorem, 3 equations.

Table of Contents

  1. Intersections in a sequence of sets of positive measure
  2. Homogenizing the sequence
  3. de Finetti's theorem
  4. Exchangeable versus spreadable sequences
  5. Model theoretic stability of probability algebras
  6. Intersecting multi-parametric families of events
  7. Structural Ramsey theory, and infinite limits of Ramsey classes
  8. Finding an "exchangeable" counterexample
  9. (Relatively) Exchangeable random structures
  10. A higher amalgamation condition on the indexing structure
  11. Presentation for random relational structures
  12. Getting rid of the ordering
  13. Open questions and future directions

Key Result

Theorem \oldthetheorem

For every finite bipartite graph $H = (V_0,W_0, E_0)$ and $\varepsilon \in (0,1]$ there exists a finite bipartite graph $G = (V,W, E)$ and $\delta > 0$ (depending only on $H$ and $\varepsilon$) satisfying the following. Assume that $(X, \mathcal{B}, \mu)$ is a probability space, and for every $(v, w

Theorems & Definitions (5)

  • Definition 1
  • Theorem \oldthetheorem: Chernikov, Towsner CheTow
  • Definition 2
  • Definition 3
  • Definition 4