  We construct a set of positive integers A in {1,..., n} with |A|>> n^{2/3}
that does not contain Hilbert cubes of dimension 3. As a consequence we prove
that ex(n; K^(3)(2,2,2))>> n^{8/3} where K^(3)(2,2,2) is the simplest complete
3-partite hypergraph. This is the first case of an improvement on the trivial
lower bound for ex(n; L) when L is a complete r-partite hypergraph.
