We prove that every partially ordered set on n elements contains k subsets \(A_{1},A_{2},\dots ,A_{k}\) such that either each of these subsets has size \(\Omega (n/k^{5})\) and, for every \(i<j\) , every element in \(A_{i}\) is less than or equal to every element in \(A_{j}\) , or each of these subsets has size \(\Omega (n/(k^{2}\log n))\) and, for every \(i \not = j\) , every element in \(A_{i}\) is incomparable with every element in \(A_{j}\) for \(i\ne j\) . This answers a question of the first author from 2006. As a corollary, we prove for each positive integer h there is \(C_h\) such that for any h partial orders \(<_{1},<_{2},\dots ,<_{h}\) on a set of n elements, there exists k subsets \(A_{1},A_{2},\dots ,A_{k}\) each of size at least \(n/(k\log n)^{C_{h}}\) such that for each partial order \(<_{\ell }\) , either \(a_{1}<_{\ell }a_{2}<_{\ell }\dots <_{\ell }a_{k}\) for any tuple of elements \((a_1,a_2,\dots ,a_k) \in A_1\times A_2\times \dots \times A_k\) , or \(a_{1}>_{\ell }a_{2}>_{\ell }\dots >_{\ell }a_{k}\) for any \((a_1,a_2,\dots ,a_k) \in A_1\times A_2\times \dots \times A_k\) , or \(a_i\) is incomparable with \(a_j\) for any \(i\ne j\) , \(a_i\in A_i\) and \(a_j\in A_j\) . This improves on a 2009 result of Pach and the first author motivated by problems in discrete geometry.
