For 10 points, what is this theorem, whose second namesake is better known for a chaos theory experiment with Fermi and Pasta and for helping Edward (*) Teller design thermonuclear weapons?
In a book about "using" this theorem, Jiri Matousek presents Alon's proof of the necklace theorem which makes use of it. Tucker's lemma is a combinatorial analog of this theorem, which Barany and Greene used to give tight lower bounds on the chromatic number of Kneser graphs. Corollaries of this theorem include the Lyusternik-Shnirel'man theorem, as well as the fact that any three solid figures in three-space can be simultaneously bisected, known as the ham sandwich theorem. This theorem implies the Brouwer fixed-point theorem, and it is often illustrated by claiming that there are two antipodal points on the Earth's surface with equal temperature and barometric pressure. For 10 points, what is this theorem, whose second namesake is better known for a chaos theory experiment with Fermi and Pasta and for helping Edward (*) Teller design thermonuclear weapons?
In a book about "using" this theorem, Jiri Matousek presents Alon's proof of the necklace theorem which makes use of it.
Borsuk-Ulam theorem [accept ham sandwich theorem before mention, even though no one's gonna say it]