The ehrenfeucht-fraïssé-game of length ω₁

Let 픘 and 픙 be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-fraïssé-game of length ω₁ of 픘 and 픙 which we denote by G_ω1,(픘, 픙). This game is like the ordinary Ehrenfeucht-fraïssé-game of L_ωω except that there are ω₁ moves. It is clear that G_ω1,(픘, 픙) is determined if 픘 and 픙 are of cardinality ≤ N₁. We prove the following results: Theorem 1. If V = L, then there are models 픘 and 픙 of cardinality N₂ such that the game G_ω1,(픘, 픙) is nondetermined. Theorem 2. If it is consistent that there is a measurable cardinal, then it is consistent that.G_ω1,(픘, 픙) is determinedf or all 픘 and 픙 of cardinality ≤ N₂. Theorem 3. For any k ≤ N₃ there are 픘 and 픙 of cardinality k such that the game G_ω1,(픘, 픙) is nondetermined.