Partition regularity

In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets.

Given a set ${\displaystyle X}$, a collection of subsets ${\displaystyle \mathbb {S} \subset {\mathcal {P}}(X)}$ 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 ${\displaystyle A\in \mathbb {S} }$, and any finite partition ${\displaystyle A=C_{1}\cup C_{2}\cup \cdots \cup C_{n}}$, there exists an i ≤ n, such that ${\displaystyle C_{i}}$ belongs to ${\displaystyle \mathbb {S} }$. Ramsey theory is sometimes characterized as the study of which collections ${\displaystyle \mathbb {S} }$ are partition regular.

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