An Ehrenfeucht-Fraisse Game for the Logic L-omega1-omega Tong Wang Abstract: The Ehrenfeucht-Fraïssé Game is very useful in studying separation and equivalence results in logic. The usual finite Ehrenfeucht-Fraïssé Game EFn characterizes separation in first order logic Lωω. The infinite Ehrenfeucht-Fraïssé Game EFω and the Dynamic Ehrenfeucht-Fraïssé Game EFDα characterize separation in L∞ω, the logic with arbitrary conjunctions and disjunctions of formulas. The logic Lω1 ω is the extension of first order logic with countable conjunctions and disjunctions of formulas. It is the most immediate, and perhaps the most important infinitary logic. However, there is no Ehrenfeucht-Fraïssé Game in the literature that characterizes separation in Lω1ω. In this thesis we introduce an Ehrenfeucht-Fraïssé Game for the logic Lω1ω . This game is based on a game for propositional and first order logic introduced by Hella and Väänänen. Unlike the usual Ehrenfeucht-Fraïssé Games which are modeled solely after the behavior of quantifiers, this new game also takes into account the behavior of boolean connectives in logic. We prove the adequacy theorem for this game. In the final part of the thesis we apply this game to prove complexity results about infinite binary strings.