We investigate maximal free sequences in the Boolean algebra \ {/}\text {fin}\), as defined by Monk :593–610, 2011). We provide some information on the general structure of these objects and we are particularly interested in the minimal cardinality of a free sequence, a cardinal characteristic of the continuum denoted \. Answering a question of Monk, we demonstrate the consistency of \. In fact, this consistency is demonstrated in the model of Shelah for \ :433–443, 1992). Our paper provides a streamlined (...) and mostly self contained presentation of this construction. (shrink)

We investigate maximal free sequences in the Boolean algebra \ {/}\text {fin}\), as defined by Monk :593–610, 2011). We provide some information on the general structure of these objects and we are particularly interested in the minimal cardinality of a free sequence, a cardinal characteristic of the continuum denoted \. Answering a question of Monk, we demonstrate the consistency of \. In fact, this consistency is demonstrated in the model of Shelah for \ :433–443, 1992). Our paper provides a streamlined (...) and mostly self contained presentation of this construction. (shrink)

We investigate maximal free sequences in the Boolean algebra \ {/}\text {fin}\), as defined by Monk :593–610, 2011). We provide some information on the general structure of these objects and we are particularly interested in the minimal cardinality of a free sequence, a cardinal characteristic of the continuum denoted \. Answering a question of Monk, we demonstrate the consistency of \. In fact, this consistency is demonstrated in the model of Shelah for \ :433–443, 1992). Our paper provides a streamlined (...) and mostly self contained presentation of this construction. (shrink)

Given a family${\cal C}$of infinite subsets of${\Bbb N}$, we study when there is a Borel function$S:2^{\Bbb N} \to 2^{\Bbb N} $such that for every infinite$x \in 2^{\Bbb N} $,$S\left \in {\Cal C}$and$S\left \subseteq x$. We show that the family of homogeneous sets as given by the Nash-Williams’ theorem admits such a Borel selector. However, we also show that the analogous result for Galvin’s lemma is not true by proving that there is an$F_\sigma $tall ideal on${\Bbb N}$without a Borel selector. The (...) proof is not constructive since it is based on complexity considerations. We construct a${\bf{\Pi }}_2^1 $tall ideal on${\Bbb N}$without a tall closed subset. (shrink)