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 is a graph homomorphism, then — no matter whether the graphs involved are finite or infinite.
Proof. Colorings are homomorphisms to complete graphs. If is a homomorphism, then so is .
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 , 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.