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Non-dominating ultrafilters. (English) Zbl 1261.03140

Summary: We show that if \({\mathrm{cov}}({\mathcal M})=\kappa\), where \(\kappa\) is a regular cardinal such that \(\forall\lambda<\kappa(2^\lambda\leq\kappa)\), then for every unbounded directed family \({\mathcal H}\) of size \(\kappa\) there is an ultrafilter \({\mathcal U}_{\mathcal H}\) such that the relativized Mathias forcing \({\mathbb M}({\mathcal U}_{\mathcal H})\) preserves the unboundedness of \({\mathcal H}\). This improves a result of M. Canjar [Proc. Am. Math. Soc. 104, No. 4, 1239–1248 (1988; Zbl 0691.03030), Theorem 10]. We discuss two instances of generic ultrafilters for which the relativized Mathias forcing preserves the unboundedness of certain unbounded families of size \(<{\mathfrak c}\).

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results

Citations:

Zbl 0691.03030
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