Partition regularity

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

Given a set $$X$$, a collection of subsets $$\mathbb{S} \subset \mathcal{P}(X)$$ 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 $$A \in \mathbb{S}$$, and any finite partition $$A = C_1 \cup C_2 \cup \cdots \cup C_n$$, there exists an i ≤ n such that $$C_i$$ belongs to $$\mathbb{S}$$. Ramsey theory is sometimes characterized as the study of which collections $$\mathbb{S}$$ 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 (i.e. the infinite pigeonhole principle.)
 * Sets with positive upper density in $$\mathbb{N}$$: the upper density $$\overline{d}(A)$$ of $$A \subset \mathbb{N}$$ is defined as $$ \overline{d}(A) = \limsup_{n \rightarrow \infty} \frac{| \{1,2,\ldots,n\} \cap A|}{n}. $$ (Szemerédi's theorem)
 * For any ultrafilter $$\mathbb{U}$$ on a set $$X$$, $$\mathbb{U}$$ is partition regular: for any $$A \in \mathbb{U}$$, if $$A = C_1 \sqcup \cdots \sqcup C_n$$, then exactly one $$C_i \in \mathbb{U}$$.
 * Sets of recurrence: a set R of integers is called a set of recurrence if for any measure-preserving transformation $$T$$ of the probability space (&Omega;, &beta;, &mu;) and $$A \in \beta$$ of positive measure there is a nonzero $$n \in R$$ so that $$\mu(A \cap T^{n}A) > 0$$.
 * 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 (Van der Waerden, 1927).
 * Let $$[A]^n$$ be the set of all n-subsets of $$A \subset \mathbb{N}$$. Let $$\mathbb{S}^n = \bigcup^{ }_{A \subset \mathbb{N}} [A]^n$$. For each n, $$\mathbb{S}^n$$ is partition regular. (Ramsey, 1930).
 * For each infinite cardinal $$\kappa$$, the collection of stationary sets of $$\kappa$$ is partition regular. More is true: if $$S$$ is stationary and $$S=\bigcup_{\alpha < \lambda} S_{\alpha}$$ for some $$\lambda < \kappa $$, then some $$S_{\alpha} $$ is stationary.
 * The collection of $$\Delta$$-sets: $$A \subset \mathbb{N}$$ is a $$\Delta$$-set if $$A$$ contains the set of differences $$\{s_m - s_n : m,n \in \mathbb{N},\, n<m \}$$ for some sequence $$\langle s_n \rangle^{\infty}_{n=1}$$.
 * The set of barriers on $$\mathbb{N}$$: call a collection $$\mathbb{B}$$ of finite subsets of $$\mathbb{N}$$ a barrier if:
 * $$\forall X,Y \in \mathbb{B}, X \not\subset Y$$ and
 * for all infinite $$I \subset \cup \mathbb{B}$$, there is some $$X \in \mathbb{B}$$ such that the elements of X are the smallest elements of I; i.e. $$X \subset I$$ and $$\forall i \in I \setminus X, \forall x \in X, x<i$$.
 * This generalizes Ramsey's theorem, as each $$[A]^n$$ is a barrier. (Nash-Williams, 1965)


 * Finite products of infinite trees (Halpern–Läuchli, 1966)
 * Piecewise syndetic sets (Brown, 1968)
 * 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 (Jon Folkman, Richard Rado, and J. Sanders, 1968).
 * (m, p, c)-sets
 * IP sets
 * MTk sets for each k, i.e. k-tuples of finite sums (Milliken–Taylor, 1975)
 * Central sets; i.e. the members of any minimal idempotent in $$\beta\mathbb{N}$$, the Stone–Čech compactification of the integers. (Furstenberg, 1981, see also Hindman, Strauss, 1998)

Diophantine equations
A Diophantine equation $$P(\mathbf{x}) = 0$$ is called partition regular if the collection of all infinite subsets of $$\N$$ containing a solution is partition regular. Rado's theorem characterises exactly which systems of linear Diophantine equations $$\mathbf{A}\mathbf{x} = \mathbf{0}$$ are partition regular. Much progress has been made recently on classifying nonlinear Diophantine equations.