As addressed in this past [question](https://mathoverflow.net/questions/17006/linear-algebra-proofs-in-combinatorics), there are many applications of linear algebra to combinatorics. What about examples of applications of exterior algebras? Part 4 [here](http://gilkalai.wordpress.com/2008/12/25/lovaszs-two-families-theorem/) is one such example. What are the heuristics for when to apply them and the intuition for how to do so?