### An analysis of Karp’s interpolation theorem and the notion of $k$-consistency property

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We continue the work of [2] and prove that for λ successor, a λ-categorical theory T in ${L}_{\kappa *,\omega}$ is μ-categorical for every μ ≤ λ which is above the $\left({2}^{LS\left(T\right)}\right)\u207a$-beth cardinal.

We assume a theory T in the logic ${L}_{\kappa \omega}$ is categorical in a cardinal λ κ, and κ is a measurable cardinal. We prove that the class of models of T of cardinality < λ (but ≥ |T|+κ) has the amalgamation property; this is a step toward understanding the character of such classes of models.

This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing previous work of the author. A cardinal κ is characterized by a Scott sentence ${\varphi}_{\mathcal{M}}$ if ${\varphi}_{\mathcal{M}}$ has a model of size κ, but no model of size κ⁺. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if ${\aleph}_{\beta}$ is characterized by a Scott sentence, then ${2}^{{\aleph}_{\beta +\beta \u2081}}$ is (homogeneously) characterized by a Scott sentence, for all 0 < β₁ < ω₁....

We study the possibilities of constructing, in ZFC without any additional assumptions, strongly equivalent non-isomorphic trees of regular power. For example, we show that there are non-isomorphic trees of power ω₂ and of height ω · ω such that for all α < ω₁· ω · ω, E has a winning strategy in the Ehrenfeucht-Fraïssé game of length α. The main tool is the notion of a club-guessing sequence.

We characterize exactly the compactness properties of the product of κ copies of the space ω with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard elements in elementary extensions. We also have results involving products of possibly uncountable regular cardinals.

This paper is a survey of results on finite variable logics in finite model theory. It focusses on the common underlying techniques that unite many such results.