A multipartite analogue of Dilworth's Theorem

We prove that every partially ordered set on n elements contains k subsets A_1,A_2,…,A_k such that either each of these subsets has size Ω(n/k^5) and, for every i_ℓa_2>_ℓ…>_ℓa_k for any (a_1,a_2,…,a_k) ∈ A_1× A_2×…× A_k, or a_i is incomparable with a_j for any i j, a_i∈ A_i and a_j∈ A_j. This improves on a 2009 result of Pach and the first author motivated by problems in discrete geometry.