Optimal shadows and ideals in submatrix orders

The main result of this article is in proving a conjecture by Sali. We obtain a Kruskal-Katona-type theorem for the poset P(N;A,B), which for a finite set N and disjoint subsets A,B subset of or equal to N is the set {F subset of or equal to N \ F boolean AND A not equal theta not equal F boolean AND B}, ordered by inclusion. Such posets are known as submatrix orders. As an application we give a solution to the problem of finding an ideal of given size and maximum weight in submatrix orders and in their duals. (C) 2001 Elsevier Science B.V. All rights reserved.