Ramsey’s theorem

Ramsey’s theorem states that a particular arrows relation,

$$\omega \to {(\omega )}_k^n$$

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 $f:{[\omega ]}^2\to \{0,1\}$ 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 $X$.