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BASES AND BOREL SELECTORS FOR TALL FAMILIES
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- Journal:
- The Journal of Symbolic Logic / Volume 84 / Issue 1 / March 2019
- Published online by Cambridge University Press:
- 30 January 2019, pp. 359-375
- Print publication:
- March 2019
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- Article
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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.
