Journal of Symbolic Logic

Cohen-Stable Families of Subsets of Integers

Milos S. Kurilic

Abstract

A maximal almost disjoint (mad) family $\mathscr{A} \subseteq [\omega]^\omega$ is Cohen-stable if and only if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-unstable. It is shown that a mad family, $\mathscr{A}$, is Cohen-unstable if and only if there is a bijection G from $\omega$ to the rationals such that the sets G[A], $A \in\mathscr{A}$ are nowhere dense. An $\aleph_0$-mad family, $\mathscr{A}$, is a mad family with the property that given any countable family $\mathscr{B} \subset [\omega]^\omega$ such that each element of $\mathscr{B}$ meets infinitely many elements of $\mathscr{A}$ in an infinite set there is an element of $\mathscr{A}$ meeting each element of $\mathscr{B}$ in an infinite set. It is shown that Cohen-stable mad families exist if and only if there exist $\aleph_0$-mad families. Either of the conditions $\mathfrak{b} = \mathfrak{c}$ or $\mathfrak{a} < cov(\mathscr{K}$) implies that there exist Cohen-stable mad families. Similar results are obtained for splitting families. For example, a splitting family, $\mathscr{S}$, is Cohen-unstable if and only if there is a bijection G from $\omega$ to the rationals such that the boundaries of the sets G[S], $S \in\mathscr{S}$ are nowhere dense. Also, Cohen-stable splitting families of cardinality $\leq \kappa$ exist if and only if $\aleph_0$-splitting families of cardinality $\leq \kappa$ exist.

Article information

Source
J. Symbolic Logic, Volume 66, Issue 1 (2001), 257-270.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183746369

Mathematical Reviews number (MathSciNet)
MR1825183

Zentralblatt MATH identifier
0981.03049

JSTOR