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 n 1 − 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 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 , B ⊂ 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 .