This question was motivated by a discussion [here](https://mathoverflow.net/questions/375792/injective-choice-function-for-infinite-complete-linear-hypergraphs) and is related to a previous question [here](https://mathoverflow.net/questions/287380/perfect-matchings-in-infinite-graphs?noredirect=1&lq=1).


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 $e\in \mathcal{L}$.



Perfect matchings in infinite regular bipartite graphs
I believe this is correct (assuming $\lambda\gt0$).


If $\lambda$ is infinite then each connected component of $G$ has $\lambda$ vertices. Since the components can be handled independently, the problem reduces to the $\kappa=\lambda$ case, which can be done by a straightforward transfinite recursion.


If $\lambda$ is a positive integer, this is a classical theorem. First, the existence of a matching of $A$ *into* $B$ follows by the usual sort of compactness argument (e.g. Tychonoff's theorem) from the fact that every finite subset of $A$ can be matched into $B$, which is a famous result of graph theory with many names. Then, givem a matching of $A$ into $B$ and a matching of $B$ into $A$, we can get a perfect matching from Banach's mapping theorem, which says: Given any two mappings $f:A\to B$ and $g:B\to A$, there are partitions $A=A\_1\cup A\_2$ and $B=B\_1\cup B\_2$ such that $f(A\_1)=B\_1$ and $g(B\_2)=A\_2$.


