A multipartite analogue of Dilworth's Theorem

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.