Intersecting sets in probability spaces and Shelah's classification
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917707087085568 |
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| author | Chernikov, Artem Towsner, Henry |
| author_facet | Chernikov, Artem Towsner, Henry |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18772 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Intersecting sets in probability spaces and Shelah's classification Chernikov, Artem Towsner, Henry Combinatorics Logic Probability 05C55, 60G09, 62E10, 03C45 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. |
| title | Intersecting sets in probability spaces and Shelah's classification |
| topic | Combinatorics Logic Probability 05C55, 60G09, 62E10, 03C45 |
| url | https://arxiv.org/abs/2406.18772 |