In earlier work by the first and second authors, the equivalence of a finite square principle $\square^{\mathrm{fin}}_{\lambda,D}$ with various model-theoretic properties of structures of size $\lambda $ and regular ultrafilters was established. In this paper we investigate the principle $\square^{\mathrm{fin}}_{\lambda,D}$—and thereby the above model-theoretic properties—at a regular cardinal. By Chang’s two-cardinal theorem, $\square^{\mathrm{fin}}_{\lambda,D}$ holds at regular cardinals for all regular filters $D$ if we assume the generalized continuum hypothesis. In this paper we prove in ZFC that, for certain regular filters (...) that we call $\mathit{doubly}^{+}$ regular, $\square^{\mathrm{fin}}_{\lambda,D}$ holds at regular cardinals, with no assumption about GCH. Thus we get new positive answers in ZFC to Open Problems 18 and 19 in Chang and Keisler’s book Model Theory. (shrink)

Let $\omega$ be the first infinite ordinal (or the set of all natural numbers) with the usual order $<$ . In § 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of $\omega$ , whose cardinality is (1) a singular strong limit cardinal, (2) a strongly inaccessible cardinal. This answers two questions in [1], modulo the assumption of supercompactness. In § 2 we construct several $\lambda$ -Archimedean ultrapowers of $\omega$ under some large cardinal (...) assumptions. For example, we show that, assuming the consistency of a measurable cardinal, there may exist a $\lambda$ -Archimedean ultrapower of $\omega$ for some uncountable cardinal $\lambda$ . This answers a question in [8], modulo the assumption of measurability. (shrink)