On Weakly Tight Families

Raghavan, Dilip ; Steprāns, Juris

Canadian journal of mathematics, Tome 64 (2012) no. 6, pp. 1378-1394

Résumé

Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $c<{{\aleph }_{\omega }}$ , we construct a weakly tight family under the hypothesis $\mathfrak{s}\le \mathfrak{b}<{{\aleph }_{\omega }}$ . The case when $\mathfrak{s}<\mathfrak{b}$ is handled in ZFC and does not require $\mathfrak{b}<{{\aleph }_{\omega }}$ , while an additional PCF type hypothesis, which holds when $\mathfrak{b}<{{\aleph }_{\omega }}$ is used to treat the case $\mathfrak{s}=\mathfrak{b}$ . The notion of a weakly tight family is a natural weakening of the well-studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hrušák and García Ferreira [8], who applied it to the Katétov order on almost disjoint families.

Détail

DOI : 10.4153/CJM-2012-017-8

Mots-clés : 03E17, 03E15, 03E35, 03E40, 03E05, 03E50, 03E65, maximal almost disjoint family, cardinal invariants