Stable marriages for infinite bipartite graphs

Question

Short and informal version: Does the stable marriage problem have a solution if there are $\kappa$ men and $\kappa$ women for any cardinal $\kappa \geq \aleph_0$?

Long and formal version: Let $\kappa$ be an infinite cardinal. For any set $X$ we set $\text{Bij}(X)$ to be the set of bijections from $X$ to itself.

We are given two "preference functions" $$P_M, P_W: \kappa \to \text{Bij}(\kappa).$$ Let $\mu:M\to W$ be a bijection (also called "marriage"). We say that $(m_0, w_0)\in \kappa\times \kappa$ is a blocking pair for the marriage $\mu$ if

1. $\mu(m_0) \neq w_0$, and
2. $\big(P_M(m_0)\big)(w_0) < \mu(m_0)$ and vice versa, $\big(P_W(w_0)\big)(m_0) < \mu^{-1}(w_0)$.

A marriage is said to be stable if there are no blocking pairs.

Given $\kappa, P_M, P_W$ as above, is there always a stable marriage?

[This looks so ugly that I'm not even sure it's the correct formalisation, feel free to correct & downvote if I got it wrong.]

Answer 1

Your formal version does not look correct. For each boy $b$, there should be a total order $\leq_b$ on the set of girls $G$ (this is the preference order for $b$) and for each girl $g$, there should be a total order $\leq^g$ on the set of boys (this is the preference order for $g$). In the special case that $G$ and $B$ are both countably infinite and each $\leq_b$ and $\leq^g$ has order type $\omega$ (this is a natural generalization of the finite case), there is indeed a Stable Marriage Theorem. See Theorem 2.1 of this paper of Cenzer and Remmel. The proof of correctness is to simply extend the Gale-Shapley algorithm to the infinite case. Theorem 2.5 of the same paper shows that as long as the preference orderings $\leq_b$ and $\leq^g$ are all well-orderings, then there is a stable matching.