Big Ramsey Degrees of Countable Ordinals
Joanna Boyland, William Gasarch, Nathan Hurtig, Robert Rust
TL;DR
The paper develops a unified, bottom-up framework to compute big Ramsey degrees for countable ordinals. By introducing coloring rules (CRs) and generalized coloring rules (GCRs), it provides explicit, exact values for $T(n,S)$ across increasingly complex ordinals, starting from $\zeta$ and proceeding through $\omega^d$, $\omega^d\cdot k$, and all $\alpha<\omega^\omega$. The authors prove $T(n,\zeta)=2^n$ and derive sharp formulas such as $T(n,\omega\cdot k)=k^n$ and $T(2,\omega^2)=4$, then extend these results to all ordinals below $\omega^\omega$ via CR/GCR counting. This approach yields precise degrees and a coherent method potentially extendable to other ordered structures, illustrating a concrete, constructive alternative to prior top-down techniques.
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. Mašulović \& Šobot (2019) showed that $T(a,ω+ω)=2^a$. From this one can obtain $T(a,ζ)=2^a$. We give a direct proof that $T(a,ζ)=2^a$. Mašulović and Šobot (2019) also showed that for all countable ordinals $α< ω^ω$, and for all $a \in N$, $T(a,α)$ is finite. We find exact value of $T(a,α)$ for all ordinals less than $ω^ω$ and all $a\in N$.