# Countable ordinals and big Ramsey degrees

## TL;DR

This paper investigates the big Ramsey degrees of finite chains within countable ordinals, establishing a precise boundary at omega for finiteness.

## Contribution

It characterizes exactly which countable ordinals have finite big Ramsey degrees, revealing a sharp cutoff at omega.

## Key findings

- Countable ordinals less than omega have finite big Ramsey degrees.
- Countable ordinals omega and above have infinite big Ramsey degrees.
- Provides a complete classification of big Ramsey degrees for finite chains in countable ordinals.

## Abstract

In this paper we consider big Ramsey degrees of finite chains in countable ordinals. We prove that a countable ordinal has finite big Ramsey degrees if and only if it is smaller than $\omega^\omega$. Big Ramsey degrees of finite chains in all other countable ordinals are infinite.