Jonathan Cancino: Ideal independent families and ultrafilters
02/06/22 09:43
Tuesday, June 7, 2022 17:15
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Jonathan Cancino (Czech Academy of Sciences)
Title: Ideal independent families and ultrafilters
Abstract: A family \(\mathscr{I}\subseteq[\omega]^\omega\) is called ideal independent if no element \(A\in\mathscr{I}\) is almost contained in the union of finitely many other elements in \(\mathscr{I}\). The ideal independence number, denoted by \(\mathfrak{s}{mm}\), is defined as the minimal cardinality of a maximal ideal independent family. We will review some results about ideal independent families and the cardinal invariant \(\mathfrak{s}{mm}\). In particular we will prove that the ultrafilter number is a lower bound for \(\mathfrak{s}{mm}\). Also, we will see that the spectrum of ideal independent families, defined as the family of all cardinalities of maximal ideal independent families, can be quite rich. If time allows, we will sketch a proof that consistently \(\mathfrak{s}{mm}<\mathfrak{a}_T\), where \(\mathfrak{a}_T\) is the minimal cardinality of a family of disjoint compact sets covering the Baire space. This is joint work with V. Fischer and C. B. Switzer.
Location: room 605, Mathematical Institute, University of Wroclaw
Speaker: Jonathan Cancino (Czech Academy of Sciences)
Title: Ideal independent families and ultrafilters
Abstract: A family \(\mathscr{I}\subseteq[\omega]^\omega\) is called ideal independent if no element \(A\in\mathscr{I}\) is almost contained in the union of finitely many other elements in \(\mathscr{I}\). The ideal independence number, denoted by \(\mathfrak{s}{mm}\), is defined as the minimal cardinality of a maximal ideal independent family. We will review some results about ideal independent families and the cardinal invariant \(\mathfrak{s}{mm}\). In particular we will prove that the ultrafilter number is a lower bound for \(\mathfrak{s}{mm}\). Also, we will see that the spectrum of ideal independent families, defined as the family of all cardinalities of maximal ideal independent families, can be quite rich. If time allows, we will sketch a proof that consistently \(\mathfrak{s}{mm}<\mathfrak{a}_T\), where \(\mathfrak{a}_T\) is the minimal cardinality of a family of disjoint compact sets covering the Baire space. This is joint work with V. Fischer and C. B. Switzer.