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
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.

Diophantine equations

A Diophantine equation is called partition regular if the collection of all infinite subsets of containing a solution is partition regular. Rado's theorem characterises exactly which systems of linear Diophantine equations are partition regular. Much progress has been made recently on classifying nonlinear Diophantine equations.