# Sh:776

- Hyttinen, T., Shelah, S., & Väänänen, J. A. (2002).
*More on the Ehrenfeucht-Fraïssé game of length \omega_1*. Fund. Math.,**175**(1), 79–96. arXiv: math/0212234 DOI: 10.4064/fm175-1-5 MR: 1971240 -
Abstract:

Let A and B be two first order structures of the same relational vocabulary L. The Ehrenfeucht-Fraı̈ssé-game of length \gamma of A and B denoted by EFG_\gamma(A,B) is defined as follows: There are two players called \forall and \exists. First \forall plays x_0 and then \exists plays y_0. After this \forall plays x_1, and \exists plays y_1, and so on. Eventually a sequence \langle(x_\beta,y_\beta):\beta< \gamma\rangle has been played. The rules of the game say that both players have to play elements of A\cup B. Moreover, if \forall plays his x_\beta in A (B), then \exists has to play his y_\beta in B (A). Thus the sequence \langle(x_\beta,y_\beta): \beta<\gamma\rangle determines a relation \pi\subseteq A\times B. Player \exists wins this round of the game if \pi is a partial isomorphism. Otherwise \forall 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\leq\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\leq\omega_3. - Current version: 2001-07-30_10 (22p) published version (18p)

Bib entry

@article{Sh:776, author = {Hyttinen, Tapani and Shelah, Saharon and V{\"a}{\"a}n{\"a}nen, Jouko A.}, title = {{More on the Ehrenfeucht-Fra\"iss\'e game of length $\omega_1$}}, journal = {Fund. Math.}, fjournal = {Fundamenta Mathematicae}, volume = {175}, number = {1}, year = {2002}, pages = {79--96}, issn = {0016-2736}, mrnumber = {1971240}, mrclass = {03C55 (03C45 03C75)}, doi = {10.4064/fm175-1-5}, note = {\href{https://arxiv.org/abs/math/0212234}{arXiv: math/0212234}}, arxiv_number = {math/0212234} }