## 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.