I've just come across [this][1] proof of the Graham-Pollak Theorem by Sundar Vishwanathan (thanks to Konrad Swanepoel's sporadic comments about it on this site), that must be called *beautiful* after its author. The paper mentions (after lemma 3) that certain other linear algebra proofs can also be replaced by similar pigeonhole principle applications. But I couldn't find any, so I thought that we should collect some similar *beautiful* proofs here. I don't mind if it also uses parity or other tricks, but linear algebra should be replaced! For example, can anyone prove the Oddtown theorem?


  [1]: http://arxiv.org/abs/1007.1553