Mad families, forcing and the Suslin Hypothesis

Abstract

Let κ be a regular cardinal and P a partial ordering preserving the regularity of κ. If P is (κ-Baire and) of density κ, then there is a mad family on κ killed in all generic extensions (if and) only if below each p P there exists a κ-sized antichain. In this case a mad family [InlineMediaObject not available: see fulltext.] on κ is killed (if and) only if there exists an injection from κ onto a dense subset of Ult(P) mapping the elements of [InlineMediaObject not available: see fulltext.] onto nowhere dense sets. If 2<κ =κ, then in each generic extension of V, in which κ is the minimal cardinal obtaining new subsets, some mad family on κ is killed or an independent subset of κ appears. Also, the κ-Suslin Hypothesis holds iff there exists a mad family [InlineMediaObject not available: see fulltext.] on κ which is killed in each generic extension containing new subsets of κ and preserving P(λ) for λ<κ. © Springer-Verlag 2004.