More on the Ehrenfencht-Fraisse game of length omega₁

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Let A and B be two first order structures of the same relational vocabulary L. The Ehrenfeucht-Fraisse-game of length gamma of A and B denoted by EFGgamma(A,B) is defined as follows: There are two players called for all and exists. First for all plays x₀ and then exists plays y₀. After this for all plays x₁, and exists plays y₁, and so on. Eventually a sequence <(xbeta,ybeta): beta<gamma> has been played. The rules of the game say that both players have to play elements of A cup B. Moreover, if for all plays his xbeta in A (B), then exists has to play his ybeta in B (A). Thus the sequence < (xbeta,ybeta):beta<gamma > determines a relation pi subseteq AxB. Player exists wins this round of the game if pi is a partial isomorphism. Otherwise for all wins. The game EFGgammaᵈelta (A,B) is defined similarly except that the players play sequences of length<delta at a time. Theorem 1: The following statements are equiconsistent relative to ZFC: (A) There is a weakly compact cardinal. (B) CH and EFomega₁(A,B) is determined for all models A,B of cardinality aleph₂. Theorem 2: Assume that 2ᵒmega <2omega₃ and T is a countable complete first order theory. Suppose that one of (i)-(iii) below holds. Then there are A,B models T of power omega₃ such that for all cardinals 1<theta<=omega₃, EFᵗhetaomega₁(A,B) is non-determined. [(i)] T is unstable. [(ii)] T is superstable with DOP or OTOP. [(iii)] T is stable and unsuperstable and 2ᵒmega <= omega₃.