Optimal pairs of incomparable clouds in multisets

Ahlswede R, Khachatrian LH. Optimal pairs of incomparable clouds in multisets. GRAPHS AND COMBINATORICS. 1996;12(1):97-137.We consider the partially ordered set ([k](n), less than or equal to), 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 subset of [k](n), that is, a not less than or equal to b, a not greater than or equal to b for all a is an element of A, b is an element of B or (in short notation) A superset of\subset of B. We are concerned with the growth of the functions f(n): {0, 1,...,k(n)} --> {0, 1,...,k(n)}, n is an element of N, defined by f(n)(alpha) = max {\B\ A, B subset of [k](n) with \A\ = alpha and A superset of\subset of 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 results by Seymour, Hilton, Ahlswede and Zhang