Big Ramsey spectra of countable chains

TL;DR

This paper investigates the big Ramsey spectra of countable scattered chains, establishing conditions under which these spectra are finite or infinite based on Hausdorff rank and sum boundedness.

Contribution

It characterizes when countable scattered chains have finite big Ramsey spectra, extending previous results on non-scattered chains.

Findings

Countable scattered chains of infinite Hausdorff rank lack finite big Ramsey spectra.

Finite Hausdorff rank with bounded finite sums implies finite big Ramsey spectra.

All non-scattered countable chains have finite big Ramsey spectra.

Abstract

A big Ramsey spectrum of a countable chain (i.e. strict linear order) C is a sequence of big Ramsey degrees of finite chains computed in C. In this paper we consider big Ramsey spectra of countable scattered chains. We prove that countable scattered chains of infinite Hausdorff rank do not have finite big Ramsey spectra, and that countable scattered chains of finite Hausdorff rank with bounded finite sums have finite big Ramsey spectra. Since big Ramsey spectra of all non-scattered countable chains are finite by results of Galvin, Laver and Devlin, in order to complete the characterization of countable chains with finite big Ramsey spectra (or degrees) one still has to resolve the remaining case of countable scattered chains of finite Hausdorff rank whose finite sums are not bounded.