A new combinatorial characterization of the minimal cardinality of a subset of R which is not of first category

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}^ωwith the property that for each f: ω-> \bigcup_{n \in ω}{0,1}^n there exists a sequence {a_n}_{n \in ω} belonging to S such that for infinitely many i \in ωthe infinite sequence {a_{i+n}}_{n \in ω} extends the finite sequence f(i). We inform that S \subseteq {0,1}^ωis not of first category if and only if for each f: ω-> \bigcup_{n \in ω}{0,1}^n there exists a sequence {a_n}_{n \in ω} belonging to S such that for infinitely many i \in ωthe infinite sequence {a_{i+n}}_{n \in ω} extends the finite sequence f(i).