BASES AND BOREL SELECTORS FOR TALL FAMILIES

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( x \right) \in {\Cal C}$and$S\left( x \right) \subseteq x$. We show that the family of homogeneous sets (with respect to a partition of a front) as given by the Nash-Williams’ theorem admits such a Borel selector. However, we also show that the analogous result for Galvin’s lemma is not true by proving that there is an$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.