Partition regularity In combinatorics , a branch of mathematics , partition regularity is one notion of largeness for a collection of sets. Given a set , a collection of subsets 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 , and any finite partition, there exists an i ≤ n , such that belongs to . Ramsey theory is sometimes characterized as the study of which collections are partition regular.Examples 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 sets with positive upper density in : the upper density of is defined as For any ultrafilter on a set, is partition regular: for any, if, then exactly one . sets of recurrence: a set R of integers is called a set of recurrence if for any measure preserving transformation of the probability space and of positive measure there is a nonzero so that. Call a subset of natural numbers a.p.-rich if it contains arbitrarily long arithmetic progressions . Then the collection of a.p.-rich subsets is partition regular. Let be the set of all n -subsets of. Let. For each n, is partition regular.. For each infinite cardinal , the collection of stationary sets of is partition regular. More is true: if is stationary and for some , then some is stationary. the collection of -sets: is a -set if contains the set of differences for some sequence. the set of barriers on : call a collection of finite subsets of a barrier if: * and * for all infinite, there is some such that the elements of X are the smallest elements of I; i.e. and. finite products of infinite trees piecewise syndetic sets 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. -sets IP sets MTk sets for each k , i.e. k -tuples of finite sums central sets; i.e. the members of any minimal idempotent in, the Stone–Čech compactification of the integers.
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