In "A proof of Vaught's conjecture for ω-stable theories," the notions of ENI-NDOP and eni-depth have been introduced, which are variants of the notions of NDOP and depth known from Shelah's classification theory. First, we show that for an ω-stable first-order complete theory, ENI-NDOP allows tree decompositions of countable models. Then we discuss the relationship between eni-depth and the complexity of the isomorphism relation for countable models of such a theory in terms of Borel reducibility as introduced by Friedman and Stanley and construct, in particular, a sequence of complete first-order ω-stable theories with increasing and cofinal eni-depth and isomorphism relations which are strictly increasing with respect to Borel reducibility.
"Comparing Borel Reducibility and Depth of an ω-Stable Theory." Notre Dame J. Formal Logic 50 (4) 365 - 380, 2009. https://doi.org/10.1215/00294527-2009-016