The complexity of isomorphism for complete theories of linear orders with unary predicates

Suppose A is a linear order, possibly with countably many unary predicates added. We classify the isomorphism relation for countable models of Th(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {Th}(A)$$\end{document} up to Borel bi-reducibility, showing there are exactly five possibilities and characterizing exactly when each can occur in simple model-theoretic terms. We show that if the language is finite (in particular, if there are no unary predicates), then the theory is ℵ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _0$$\end{document}-categorical or Borel complete; this generalizes a theorem due to Schirmann (Theories des ordres totaux et relations dequivalence. Master’s thesis, Universite de Paris VII, 1997).