  In this note we collect some known information and prove new results about
the small uncountable cardinal $\mathfrak q_0$. The cardinal $\mathfrak q_0$ is
defined as the smallest cardinality $|A|$ of a subset $A\subset \mathbb R$
which is not a $Q$-set (a subspace $A\subset\mathbb R$ is called a $Q$-set if
each subset $B\subset A$ is of type $F_\sigma$ in $A$). We present a simple
proof of a folklore fact that $\mathfrak p\le\mathfrak q_0\le\min\{\mathfrak
b,\mathrm{non}(\mathcal N),\log(\mathfrak c^+)\}$, and also establish the
consistency of a number of strict inequalities between the cardinal $\mathfrak
q_0$ 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
$\mathfrak{p} < \mathfrak{lr} < \mathfrak{q}_0$, where $\mathfrak{lr}$ denotes
the linear refinement number. Another new result is the upper bound $\mathfrak
q_0\le\mathrm{non}(\mathcal I)$ holding for any $\mathfrak q_0$-flexible cccc
$\sigma$-ideal $\mathcal I$ on $\mathbb R$.
