Big Ramsey Degrees of Countable Ordinals

Abstract

Ramsey's theorem states that for all finite colorings of an infinite set, there exists an infinite homogeneous subset. What if we seek a homogeneous subset that is also order-equivalent to the original set? Let $S$ be a linearly ordered set and $a \in N$. The big Ramsey degree of $a$ in $S$, denoted $T(a,S)$, is the least integer $t$ such that, for any finite coloring of the $a$-subsets of $S$, there exists $S'\subseteq S$ such that (i) $S'$ is order-equivalent to $S$, and (ii) if the coloring is restricted to the $a$-subsets of $S'$ then at most $t$ colors are used. Ma\v{s}ulovi\'{c} \& \v{S}obot (2019) showed that $T(a,\omega+\omega)=2^a$. From this one can obtain $T(a,\zeta)=2^a$. We give a direct proof that $T(a,\zeta)=2^a$. Ma\v{s}ulovi\'{c} and \v{S}obot (2019) also showed that for all countable ordinals $\alpha < \omega^\omega$, and for all $a \in N$, $T(a,\alpha)$ is finite. We find exact value of $T(a,\alpha)$ for all ordinals less than $\omega^\omega$ and all $a\in N$.