Submission date: 26 June 2024

Abstract:

6 pages; to appear in the Proceedings of the 14th Panhellenic Logic Symposium.

For n ∈ ℕ and ε > 0, given a sufficiently long sequence of events in a probability space all of measure at least ε, 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(A_i ∩ B_j) ≤ p and μ( A_j ∩ B_i) ≥ 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.

Mathematics Subject Classification: 05C55, 60G09, 62E10, 03C45

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Full text arXiv 2406.18772: pdf, ps.