We say two spanning trees of a graph are completely independent if their edge sets are disjoint, and for each pair of vertices, the paths between them in each spanning tree do not have any other vertex in common. Pai and Chang constructed two such spanning trees in the hypercube Q_n for sufficiently large n, while Kandekar and Mane recently showed there are 3 pairwise completely independent spanning trees in hypercubes Q_n for sufficiently large n. We prove that for each k, there exist k completely independent spanning trees in Q_n for sufficiently large n. In fact, we show that there are (1/12+o(1))n spanning trees in Q_n.
