In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the continuum and combinatorics on successors of singular cardinals.

Write ${\displaystyle \kappa ,\lambda }$ for ordinals, ${\displaystyle m}$ for a cardinal number (finite or infinite) and ${\displaystyle n}$ for a natural number. Erdős & Rado (1956) introduced the notation

as a shorthand way of saying that every partition of the set ${\displaystyle [\kappa ]^{n}}$ of ${\displaystyle n}$-element subsets of ${\displaystyle \kappa }$ into ${\displaystyle m}$ pieces has a homogeneous set of order type ${\displaystyle \lambda }$. A homogeneous set is in this case a subset of ${\displaystyle \kappa }$ such that every ${\displaystyle n}$-element subset is in the same element of the partition. When ${\displaystyle m}$ is 2 it is often omitted. Such statements are known as partition relations.

Assuming the axiom of choice, there are no ordinals ${\displaystyle \kappa }$ with ${\displaystyle \kappa \rightarrow (\omega )^{\omega }}$, so ${\displaystyle n}$ is usually taken to be finite. An extension where ${\displaystyle n}$ is almost allowed to be infinite is the notation

which is a shorthand way of saying that every partition of the set of finite subsets of ${\displaystyle \kappa }$ into ${\displaystyle m}$ pieces has a subset of order type ${\displaystyle \lambda }$ such that for any finite ${\displaystyle n}$, all subsets of size ${\displaystyle n}$ are in the same element of the partition. When ${\displaystyle m}$ is 2 it is often omitted.

Another variation is the notation

which is a shorthand way of saying that every coloring of the set ${\displaystyle [\kappa ]^{n}}$ of ${\displaystyle n}$-element subsets of ${\displaystyle \kappa }$ with 2 colors has a subset of order type ${\displaystyle \lambda }$ such that all elements of ${\displaystyle [\lambda ]^{n}}$ have the first color, or a subset of order type ${\displaystyle \mu }$ such that all elements of ${\displaystyle [\mu ]^{n}}$ have the second color. A coloring of ${\displaystyle [\kappa ]^{n}}$is a function ${\displaystyle f:[\kappa ]^{n}\rightarrow p}$.

Some properties of this include: (in what follows ${\displaystyle \kappa }$ is a cardinal)