On disjoint Borel uniformizations (Q1805831)

In 1973, D. G. Larmann showed that any closed subset of the plane with uncountable vertical cross-sections has \(\aleph_1\) disjoint Borel uniformizing sets. In the paper under review it is shown that Larmann's result is best possible: there exist closed sets with uncountable cross-sections which do not have more than \(\aleph_1\) disjoint Borel uniformizations, even if the continuum is much larger than \(\aleph_1\). This also answers some questions of Mauldin in the negative. The proof is based on a result of Stern, which uses Steel' method of forcing with tagged trees. In the present paper, a full presentation of this method, written in terms of Baire Category rather than forcing, is given.