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In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets.
Given a set X {\displaystyle X} , a collection of subsets S ⊂ P {\displaystyle \mathbb {S} \subset {\mathcal {P}}} is called partition regular if every set A in the collection has the property that, no matter how A is partitioned into finitely many subsets, at least one of the subsets will also belong to the collection. That is,for any A ∈ S {\displaystyle A\in \mathbb {S} } , and any finite partition A = C 1 ∪ C 2 ∪ ⋯ ∪ C n {\displaystyle A=C_{1}\cup C_{2}\cup \cdots \cup C_{n}} , there exists an i ≤ n, such that C i {\displaystyle C_{i}} belongs to S {\displaystyle \mathbb {S} }. Ramsey theory is sometimes characterized as the study of which collections S {\displaystyle \mathbb {S} } are partition regular.