Partition regularity

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
• MT^k 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.