We get consistency results on I(λ, T, T) under the assumption that D(T) has cardinality > ∣T∣. We get positive results and consistency results on IE(λ, T1, T).
The interest is model-theoretic, but the content is mostly set-theoretic: in Theorems 1–3, combinatorial; in Theorems 4–7 and 11(2), to prove consistency of counterexamples we concentrate on forcing arguments; and in Theorems 8–10 and 11(1), combinatorics for counterexamples; the rest are discussion and problems. In particular:
(A) By Theorems 1 and 2, if T ⊆ T1 are first order countable, T complete stable but ℵ0-unstable, λ > ℵ0, and ∣D(T)∣ > ℵ0, then IE(λ, T1, T) > Min{2λ, ℶ2}.
(B) By Theorems 4,5,6 of this paper, if e.g. V = L, then in some generic extension of V not collapsing cardinals, for some first order T ⊆ T1, ∣T∣ = ℵ0, ∣T1∣ = ℵ1, ∣D(T)∣ = ℵ2 and IE(ℵ2, T1, T) = 1.
This paper (specifically the ZFC results) is continued in the very interesting work of Baldwin on diversity classes [Bl]. Some more advances can be found in the new version of [Sh300] (see Chapter III, mainly §7); they confirm 0.1, 0.2 and 14(1), 14(2).