A multipartite analogue of Dilworth's Theorem
Jacob Fox, Huy Tuan Pham
TL;DR
This work provides a multipartite extension of Dilworth's theorem by showing that any poset on n elements contains k disjoint blocks A_1,...,A_k that are either chained A_1>...>A_k with substantial block size or pairwise totally incomparable with substantial block size. It introduces a general framework with functions f and g to obtain a trade-off between the two outcomes, recovering concrete bounds |A_i| ≥ Θ(n/k^5) in the chain case and |A_i| ≥ Θ(n/(k^2 log n)) in the incomparability case; it also proves these bounds are tight up to constants in the n-dependency. The results extend to h partial orders, guaranteeing homogeneous blocks across all orders with quantitative size |A_j| ≥ n/(10k log n)^{12^{h+1}}, via the cake-cutting lemma and algorithmic constructions (Condense/Select). These findings improve on prior work and have geometric consequences for Erdős-Hajnal-type properties in intersection graphs and related geometries, as well as practical algorithmic implications for finding such structures.
Abstract
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 $Ω(n/k^{5})$ 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 $Ω(n/(k^{2}\log n))$ 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.