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.

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