High Quality Content by WIKIPEDIA articles! In mathematics, the notion of partition regularity in combinatorics is one approach to explaining when a set system is quite large. The collection of all infinite subsets of an infinite set X is a prototypical example. In this case partition regularity asserts that every finite partition of an infinite set has an infinite cell (i.e. the infinite pigeonhole principle.) Call a subset of natural numbers i.p.-rich if it contains arbitrarily large finite sets together with all their finite sums. Then the collection of i.p.-rich subsets is partition regular (Folkman–Rado–Sanders, 1968).

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