This question was motivated by a discussion here and is related to a previous question here.
Let $\kappa$ and $\lambda$ be cardinals such that $0<\lambda\leq \kappa$. Let $G=(A\cup B, E)$ be a bipartite graph with $|A|=\kappa=|B|$ such that every vertex has degree $\lambda$. Does $G$ have a perfect matching?
If so, it would in particular imply that a (non-degenerate) projective plane $(\mathcal{P}, \mathcal{L})$ has a bijection $f:\mathcal{L}\to \mathcal{P}$ such that $f(e)\in e$ for all $e\in \mathcal{L}$ without having to first prove that if $|\mathcal{P}|$ is infinite then $|e|=|\mathcal{P}|$ for all $e\in \mathcal{L}$.