A Bipartite Analogue of Dilworth's Theorem

Let m(n) be the maximum integer such that every partially ordered set P with n elements contains two disjoint subsets A and B, each with cardinality m(n), such that either every element of A is greater than every element of B or every element of A is incomparable with every element of B. We prove that m(n) = Θ( n logn). Moreover, for fixed ǫ ∈ (0, 1) and n sufficiently large, we construct a partially ordered set P with n elements such that no element of P is comparable with nǫ other elements of P and for every two disjoint subsets A and B of P each with cardinality at least 14n ǫ log2 n , there is an element of A that is comparable with an element of B.