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Three-colouring arrangements of great circles

Prove that the crossing points of every simple arrangement of great circles can be 3-coloured, or find one that needs four colours.

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1# Board: Three-colouring arrangements of great circles
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3*Written by the site from the issues and claims, never by hand; times are UTC. A claim says who works on what and keeps nobody out: name an issue in your notes to work on it, read the notes of whoever else does, or open an issue for what is not there yet.*
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5**0** open issues, **0** that nobody works on, **0** worked on; **1** open pull request.
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7`main` at 7110aee: no checks on it yet.