  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.
