Ordinal Partitions

The theory of partition relations concerns generalisations of Ramsey's theorem. For any ordinal $\alpha$, write $\alpha \to (\alpha ,m{)}^2$ if for each function $f$ from unordered pairs of elements of $\alpha$ into $\{0,1\}$, either there is a subset $X\subseteq \alpha$ order-isomorphic to $\alpha$ such that $f\{x,y\}=0$ for all $\{x,y\}\subseteq X$, or there is an $m$ element set $Y\subseteq \alpha$ such that $f\{x,y\}=1$ for all $\{x,y\}\subseteq Y$. (In both cases, with $\{x,y\}$ we require $x\ne y$.) In particular, the infinite Ramsey theorem can be written in this notation as $\omega \to (\omega ,\omega {)}^2$, or if we restrict $m$ to the positive integers as above, then $\omega \to (\omega ,m{)}^2$ for all $m$. This entry formalises Larson's proof of ${\omega }^{\omega }\to ({\omega }^{\omega },m{)}^2$ along with a similar proof of a result due to Specker: ${\omega }^2\to ({\omega }^2,m{)}^2$. Also proved is a necessary result by Erdős and Milner: ${\omega }^{1+\alpha \cdot n}\to ({\omega }^{1+\alpha },2^n{)}^2$.