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Partition property

A partition property is an infinitary combinatorical principle in set theory. Partition properties are best associated with various large cardinal axioms (all of which are below measurable), but can also be associated with infinite graphs.

The pigeonhole principle famously states that if there are $n$ pigeons in $n-m$ holes, then at least one hole contains at least $m$ pigeons. Partition properties are best motivated as generalizations of the pigeonhole principle to infinite cardinals. For example, if there are $\aleph_1$ pigeons and there are $\aleph_0$ many holes, then at least one hole contains $\aleph_1$ pigeons.


There are quite a few definitions involved with partition properties. In fact, partition calculus, the study of partition properties, almost completely either comprisse or describes most of infinitary combinatorics itself, so it would make sense that the terminology involved with it is quite unique.

Square Bracket Notation

The square bracket notation is somewhat simple and easy to grasp (and used in many other places). Let $X$ be a set of ordinals. $[X]^\beta$ for some ordinal $\beta$ is the set of all subsets $x\subseteq X$ such that $(x,<)$ has order-type $\beta$; that is, there is a bijection $f$ from $x$ to $\beta$ such that $f(a)<f(b)$ iff $a<b$ for each $a$ and $b$ in $x$. Such a bijection is often called an order-isomorphism.

$[X]^{<\beta}$ for some ordinal $\beta$ is simply defined as the union of all $[X]^{\alpha}$ for $\alpha<\beta$, the set of all subsets $x\subseteq X$ with order-type less than $\beta$. In the case of $\omega$, $[X]^{<\omega}$ is the set of all finite subsets of $X$.

Homogeneous Sets

Let $f:[\kappa]^\beta\rightarrow\lambda$ be a function (in this study, such functions are often called partitions). A set $H\subseteq\kappa$ is then called homogeneous for $f$ when for any two subsets $h_0,h_1\subseteq H$ of order type $\beta$, $f(h_0)=f(h_1)$. This is equivalent to $f$ being constant on $[H]^\beta$.

In another case, let $f:[\kappa]^{<\omega}\rightarrow\lambda$ be a function. A set $H\subseteq\kappa$ is then called homogeneous for $f$ when for any two finite subsets $h_0,h_1\subseteq H$ of the same size, $f(h_0)=f(h_1)$.

The Various Partition Properties

Let $\kappa$ and $\lambda$ be cardinals and let $\alpha$ and $\beta$ be ordinals. Then, the following notations are used for the partition properties:

Let $\nu$ be a cardinal. The square bracket partition properties are defined as follows:

Theorems and Large Cardinal Axioms

There are several theorems in the study of partition calculus. Namely:

In terms of large cardinal axioms, many can be described using a partition property. Here are those which can be found on this website:


  1. Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag.
  2. Jech, T. J. (2003). Set Theory (Third). Springer-Verlag.
  3. Drake, F. (1974). Set Theory: An Introduction to Large Cardinals. North-Holland Pub. Co.
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