  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
$\mu \left( A_i \cap B_j \right) \leq p$ and $\mu \left( A_j \cap B_i\right)
\geq q$ 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.
