### 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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