Optimal pairs of incomparable clouds in multisets

We consider the partially ordered set ([k]", <), which is defined as n-th product of the chain [k] = {0,1, 2 . . . . . k 1}, and study pairs (A, B) of incomparable sets A, B ~ [k]", that is, a :g b, a :~ b for all a ~ A, b ~ B or (in short notation) A : :~ B. We are concerned with the growth of the functions f,: {0, 1 . . . . . k ~} ---, {0, 1 . . . . . k"}, n ~ N, defined by f,(ct) = max{IN: A,B c [k] * with Iat = ~ and A ~ B} and a characterisation of pairs (A, B), which assume this bound. In the previously studied case k = 2 our results are considerably sharper than earlier result s by Seymour, Hilton, Ahlswede and Zhang.