Our "long term and large scale" aim is to characterize the first order theories T (at least the countable ones) such that for every ordinal α there are λ, M₁, M₂ such that M₁ and M₂ are non-isomorphic models of T of cardinality λ which are EF⁺ _α,λ-equivalent. We expect that as in the main gap [11, XII], we get a strong dichotomy, i.e., on the non-structure side we have stronger, better examples, and on the structure side we have an analogue of [11, XIII]. We presently prove the consistency of the non-structure side for T which is χ₀-independent (= not strongly dependent), even for PC(T₁,T).

Tbilisi Mathematical Journal, Vol. 1(2008), pp. 133-164