  The Carlson-Simpson lemma is a combinatorial statement occurring in the proof
of the Dual Ramsey theorem. Formulated in terms of variable words, it
informally asserts that given any finite coloring of the strings, there is an
infinite sequence with infinitely many variables such that for every valuation,
some specific set of initial segments is homogeneous. Friedman, Simpson, and
Montalban asked about its reverse mathematical strength. We study the
computability-theoretic properties and the reverse mathematics of this
statement, and relate it to the finite union theorem. In particular, we prove
the Ordered Variable word for binary strings in ACA0.
