  In this note we establish a Ramsey-type result for certain subsets of the
$n$-dimensional cube. This can then be applied to obtain reasonable bounds on
various related structures, such as (partial) Hales-Jewett lines for alphabets
of sized 3 and 4, Hilbert cubes in sets of real numbers with small sumsets,
"corner" in the integer lattice in the plane, and 3-term geometric progressions
in integers.
