Ramsey's theorem

Ramsey's theorem states that a particular arrows relation,

$$\omega\rightarrow(\omega)^n_k$$

holds for any integers $n$ and $k$ .

In words, if $f$ is a function on sets of integers of size $n$ whose range is finite then there is some infinite $X\subseteq\omega$ such that $f$ is constant on the subsets of $X$ of size $n$ .

As an example, consider the case where $n=k=2$ , and $f\colon [\omega]^2\rightarrow\{0,1\}$ is defined by:

Then let $X\subseteq\omega$ be the set of integers which are not perfect squares. This is clearly infinite, and obviously if $x,y\in X$ then neither $x=y^2$ nor $y=x^2$ , so $f$ is homogeneous on $X$ .