Searching for cofinal subsets of directed sets subject to finite constraints

Question

Let $(P,\leq)$ be a directed set with uncountable cofinality. For every element $p\in P$, we are given a finite set $c_p\subset P\smallsetminus \{p\}$ of "incompatible elements". We say that a subset $Q\subseteq P$ is compatible if for every $q\in Q$ we have $c_q\cap Q = \emptyset$.

Question: is it always possible to find a cofinal compatible subset?

Remarks:

• I can show that the answer is yes if there is a finite uniform upper bound on the cardinalities of the sets $c_p$.

• It was pointed out to me that the assumption that the cofinality be uncountable is certainly necessary, as seen by considering $(\mathbb{N},\leq)$ with $c_n = \{1,2,...,n-1\}$.

• I suspect that a (positive) solution would require some argument in infinite combinatorics.

• I am very happy to assume the axiom of choice (or any other useful set of axioms).

Answer 1

No, it is not always possible. For a counterexample, let $P$ be the set of all finite subsets of some uncountable set $X$, ordered by inclusion (that is, $a \leq b \Leftrightarrow a \subseteq b$). In this poset, a subset $D$ of $P$ is cofinal if and only if for every finite subset $a$ of $X$, there is some $b \in D$ with $b \supseteq a$.

For each $a \in P$, let $c_a$ be the (finite) set of all proper subsets of $a$. Suppose $D$ is cofinal in $P$, and fix some $a \in D$. There is some $x \in X \setminus a$, and (because $D$ is cofinal) there is some $b \in D$ with $b \supseteq a \cup \{x\}$. But now we have $b \in D$ and $a \in D$ and $a \in c_b$.