More on the Ehrenfeucht-Fraïssé game of length ω₁

By results of [9] there are models Θ and ℬ for which the Ehrenfeucht-Fraïssé game of length ω₁, EFG_ω1 (Θ, ℬ), is non-determined, but it is consistent relative to the consistency of a measurable cardinal that no such models have cardinality ≤ N₂. We now improve the work of [9] in two ways. Firstly, we prove that the consistency strength of the statement "CH and EFG_ω1 (Θ, ℬ) is determined for all models Θ and ℬ of cardinality N₂" is that of a weakly compact cardinal. On the other hand, we show that if 2^N0 ≤ 2^N3, T is a countable complete first order theory, and one of (i) T is unstable, (ii) T is superstable with DOP or OTOP, (iii) T is stable and unsuperstable and 2^N0 ≤ ^N3, holds, then there are A, B |= T of power N₃ such that EFG_ω1 (A, B) is non-determined.