Math facts with simple proofs that are still (somewhat) surprising

Sometimes it happens to me that I become aware of a mathematical fact that surprises me – and the proof for the fact would even fit in a tweet.

When goofing around with graph homomorphisms, I realised that the following is true:

Fact. If f:G\to H is a graph homomorphism, then \chi(G) \leq \chi(H) — no matter whether the graphs involved are finite or infinite.

Proof. Colorings are homomorphisms to complete graphs. If c: H\to K_{\chi(K)} is a homomorphism, then so is c \circ f : G \to K_{\chi(K)}.

The funny thing is: If I had been told just the statement of the fact “out of the blue”, I would have said that you probably need some condition on the homomorphism f:G\to H, like it being surjective etc. So I find it a bit surprising that the statement holds in this generality — even if there is a tweet-length proof for it.

About dominiczypen

I'm interested in general topology, order theory, and graph theory. This link takes you to my preprints on arXiv.
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