More on the Ehrenfeucht-Fra{\"\i}ss\'e game of length $\omega_1$

by Hyttinen and Shelah and Vaananen. [HShV:776]
Fundamenta Math, 2002
Let A and B be two first order structures of the same relational vocabulary L . The Ehrenfeucht-Fra{i}sse-game of length gamma of A and B denoted by EFG_gamma (A,B) is defined as follows: There are two players called for all and exists . First for all plays x_0 and then exists plays y_0 . After this for all plays x_1, and exists plays y_1, and so on. Eventually a sequence <(x_beta,y_beta): 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 x_beta in A (B), then exists has to play his y_beta in B (A). Thus the sequence <(x_beta,y_beta): beta < gamma > determines a relation pi subseteq A x B . Player exists wins this round of the game if pi is a partial isomorphism. Otherwise for all wins. The game EFG_gamma^delta (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 EF_{omega_1}(A,B) is determined for all models A,B of cardinality aleph_2 . Theorem 2: Assume that 2^omega <2^{omega_3} 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 omega_3 such that for all cardinals 1< theta <= omega_3, EF^theta_{omega_1}(A,B) is non-determined. [(i)] T is unstable. [(ii)] T is superstable with DOP or OTOP. [(iii)] T is stable and unsuperstable and 2^omega <= omega_3 .

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