## Journal of Symbolic Logic

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

### Cohen-Stable Families of Subsets of Integers

#### 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

**Permanent link to this document**

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

**Mathematical Reviews number (MathSciNet)**

MR1825183

**Zentralblatt MATH identifier**

0981.03049

**JSTOR**

links.jstor.org

**Subjects**

Primary: 03E05: Other combinatorial set theory

Secondary: 03E35: Consistency and independence results 03E40: Other aspects of forcing and Boolean-valued models

**Keywords**

Cohen Forcing Mad Families Splitting Families

#### Citation

Kurilic, Milos S. Cohen-Stable Families of Subsets of Integers. J. Symbolic Logic 66 (2001), no. 1, 257--270. https://projecteuclid.org/euclid.jsl/1183746369