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,B ⊂ P , each with at least n1−o(1) elements, such that one of the following two conditions is satisfied: (1) there is an i (1 ≤ i ≤ r) such that every element of A is larger than any 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,B ⊂ C, each with at least n1−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.