  Let M denote the ideal of first category subsets of R. We prove that min{card
X: X \subseteq R, X \not\in M} is the smallest cardinality of a family S
\subseteq {0,1}^\omega with the property that for each f: \omega -> \bigcup_{n
\in \omega}{0,1}^n there exists a sequence {a_n}_{n \in \omega} belonging to S
such that for infinitely many i \in \omega the infinite sequence {a_{i+n}}_{n
\in \omega} extends the finite sequence f(i).
  We inform that S \subseteq {0,1}^\omega is not of first category if and only
if for each f: \omega -> \bigcup_{n \in \omega}{0,1}^n there exists a sequence
{a_n}_{n \in \omega} belonging to S such that for infinitely many i \in \omega
the infinite sequence {a_{i+n}}_{n \in \omega} extends the finite sequence
f(i).
