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 (Ω, β, μ) 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 $\langle s_{n}\rangle _{n=1}^{\infty }$.
• 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
• MT^k 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 $\mathbb {N}$ 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.

Further reading

• Vitaly Bergelson, N. Hindman Partition regular structures contained in large sets are abundant J. Comb. Theory A 93 (2001), 18–36.