  If we 2-color the vertices of a large hypercube what monochromatic
substructures are we guaranteed to find? Call a set S of vertices from Q_d, the
d-dimensional hypercube, Ramsey if any 2-coloring of the vertices of Q_n, for n
sufficiently large, contains a monochromatic copy of S. Ramsey's theorem tells
us that for any r \geq 1 every 2-coloring of a sufficiently large r-uniform
hypergraph will contain a large monochromatic clique (a complete
subhypergraph): hence any set of vertices from Q_d that all have the same
weight is Ramsey. A natural question to ask is: which sets S corresponding to
unions of cliques of different weights from Q_d are Ramsey?
  The answer to this question depends on the number of cliques involved. In
particular we determine which unions of 2 or 3 cliques are Ramsey and then
show, using a probabilistic argument, that any non-trivial union of 39 or more
cliques of different weights cannot be Ramsey.
  A key tool is a lemma which reduces questions concerning monochromatic
configurations in the hypercube to questions about monochromatic translates of
sets of integers.
