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:

$$f(\{x,y\})=\left\{ \begin{array}{ll} 1&\text{ if ~} x=y^2 \text{ or ~} y=x^2\\ 0&\text{otherwise} \end{array}\right.$$

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 $X$