In the paper [Hall's theorem for hypergraphs (Aharoni and Haxell, 2000)](https://onlinelibrary.wiley.com/doi/abs/10.1002/1097-0118(200010)35:2%3C83::AID-JGT2%3E3.0.CO;2-V),
the authors prove a theorem on the existence of perfect matchings in bipartite hypergraphs, using Sperner's lemma. At the last page (6), they say that "we have here a topological proof of Hall's theorem" (for bipartite graphs). I thought it should be easy to write this proof explicitly, since a simple bipartite graph is just a bipartite hypergraph in which each hyperedge is of size 2. But there is a problem: during the proof, the authors assume that the sets of neighbors (i.e., the sets $N(x)$ for each vertex $x\in X$, where $X$ is one part of the hypergraph) are pairwise-disjoint. For a hypergraph, this assumption is without loss of generality, since we can add dummy vertices to edges, and this does not affect the theorem conditions or concludion. But in a graph, we cannot add vertices to edges.


So my question is: is there an explicitly-written proof of Hall's marriage theorem, using Sperner's lemma (or a similar topological theorem)?



Proving Hall's marriage theorem using Sperner's lemma
Penny Haxell's 2011 paper [On Forming Committees](https://www.jstor.org/stable/10.4169/amer.math.monthly.118.09.777) in the American Mathematical Monthly explicitly uses Sperner's lemma to prove Hall's theorem for bipartite graphs (see theorem 4.1 and 4.2).


