A Bipartite Analogue of Dilworth's Theorem

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.

• Type: Journal
• Year: 2006
• Where: ORDER
• Authors: Jacob Fox
• Keywords: Convex Compact Sets, Fixed Positive Integer, ORDER 2006, Partial Orders