We say that a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ has *property ${\bf B}$* if there is $S\subseteq V$ such that for all $e\in E$ with $|e|>1$ we have $S\cap e \neq \emptyset \neq e \setminus S$.


Let $(P,\leq)$ be a [partially-ordered set (poset)](https://en.wikipedia.org/wiki/Partially_ordered_set). For $p\in P$, let $d(p) = \{x\in P: x\leq p\}$. We assign to $(P,\leq)$ the *principal downset hypergraph* $H\_P = \big(P, D(P)\big)$ where $D(P) = \{d(p):p\in P\}$.


**Question.** Is it consistent with ${\sf (ZF)}$ that there is a poset $(P,\leq)$ such that $H\_P$ does not have property ${\bf B}$? And is the opposite also consistent?


(In other words: no matter how you color the elements of $P$ with "red" and "blue", you will get a principal downset $d(p)$ with more than $1$ element such that every element of $d(p)$ has the same color.)



Posets such that the collection of principal down-sets does not have property ${\bf B}$
Let $M$ be the ordered Mostowski model (T. Jech, *The Axiom of Choice*, Section 4.5). Its set of atoms, $A$, has a linear order $\prec$ that makes it isomorphic to the rationals. Let $S\in M$ be a subset of $A$; then $S$ is a union of finitely many intervals. So if $S$ has a lower bound then $S$ is disjoint from (many) sets of the form $d(a)$, and if it does not have a lower bound then it contains (many) sets of the form $d(a)$. So in this model we have a linearly ordered set, $A$, such that $H\_A$ does not have property $B$.


The Jech-Sochor embedding theorem (Theorem 6.1 in Jech's book) can be used to transfer this to a model of $\mathsf{ZF}$.


