A bipartite analogue of Dilworth's theorem for multiple partial orders

Let r be a fixed positive integer. It is shown that, given any partial orders <"1, ..., <"r on the same n-element set P, there exist disjoint subsets A,[email protected]?P, each with at least n^1^-^o^(^1^) elements, such that one of the following two conditions is satisfied: (1) there is an i([email protected][email protected]?r) such that every element of A is larger than every element of B in the partial order <"i, or (2) no element of A is comparable with any element of B in any of the partial orders <"1, ..., <"r. As a corollary, we obtain that any family C of n convex compact sets in the plane has two disjoint subfamilies A,[email protected]?C, each with at least n^1^-^o^(^1^) members, such that either every member of A intersects all members of B, or no member of A intersects any member of B.