Ron Aharoni and Penny Haxell have applied Sperner's Lemma to simplicial complexes associated with families of hypergraphs to prove various results on independent transversals.  In particular, they obtain a topological proof of Hall's theorem.  Here's [a paper][1], and [an overview on Gil Kalai's blog][2].


  [1]: http://onlinelibrary.wiley.com/doi/10.1002/1097-0118%28200010%2935:2%3C83::AID-JGT2%3E3.0.CO;2-V/abstract
  [2]: http://gilkalai.wordpress.com/2012/11/25/happy-birthday-ron-aharoni/