## Regular ultrapowers at regular cardinals.(English)Zbl 1334.03043

Summary: In earlier work by the first and second authors, the equivalence of a finite square principle $$\square_{\lambda,D}^{\mathrm{fin}}$$ with various model-theoretic properties of structures of size $$\lambda$$ and regular ultrafilters was established. In this paper we investigate the principle $$\square_{\lambda,D}^{\mathrm{fin}}$$ – and thereby the above model-theoretic properties – at a regular cardinal. By Chang’s two-cardinal theorem, $$\square_{\lambda,D}^{\mathrm{fin}}$$ holds at regular cardinals for all regular filters $$D$$ if we assume the generalized continuum hypothesis (GCH). In this paper we prove in ZFC that, for certain regular filters that we call doubly$$^+$$ regular, $$\square_{\lambda,D}^{\mathrm{fin}}$$ holds at regular cardinals, with no assumption about GCH. Thus we get new positive answers in ZFC to Open Problems 18 and 19 in [C. C. Chang and H. J. Keisler, Model theory. 3rd rev. ed. Amsterdam etc.: North-Holland (1990; Zbl 0697.03022)].

### MSC:

 03E05 Other combinatorial set theory 03C55 Set-theoretic model theory 03C20 Ultraproducts and related constructions

Zbl 0697.03022
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