Decomposing complete graphs into cubes

This talk concerns when the complete graph on n vertices can be decomposed into d-dimensional cubes, where d is odd and n is even. (All other cases have been settled.) Necessary conditions are that n ≡ 1 (mod d) and n ≡ 0 (mod 2). These are known to be sufficient for d equal to 3 or 5, but for larger values of d a decomposition has been proven to exist for only sparce sets of n. We prove that for each odd d there is an infinite arithmetic progression of even integers n for which a decomposition exists