In this note we collect some known information and prove new results about
the small uncountable cardinal q0β. The cardinal q0β is
defined as the smallest cardinality β£Aβ£ of a subset AβR
which is not a Q-set (a subspace AβR is called a Q-set if
each subset BβA is of type FΟβ in A). We present a simple
proof of a folklore fact that pβ€q0ββ€min{b,non(N),log(c+)}, and also establish the
consistency of a number of strict inequalities between the cardinal q0β and other standard small uncountable cardinals. This is done by combining
some known forcing results. A new result of the paper is the consistency of
p<lr<q0β, where lr denotes
the linear refinement number. Another new result is the upper bound q0ββ€non(I) holding for any q0β-flexible cccc
Ο-ideal I on R.Comment: 8 page