A Ramsey-type result for the hypercube

We prove that for every fixed k and ` ≥ 5 and for sufficiently large n, every edge coloring of the hypercube Qn with k colors contains a monochromatic cycle of length 2`. This answers an open question of Chung. Our techniques provide also a characterization of all subgraphs H of the hypercube which are Ramsey, i.e., have the property that for every k, any k-edge coloring of a sufficiently large Qn contains a monochromatic copy of H .