# Vsevolod Voronov: On the chromatic number of 2-dimensional spheres

## Description of video

 Date: 4/1/22 Speaker : Voronov Vsevolod

## Keywords

Consider the problem of the chromatic number of a two-dimensional sphere, namely, the smallest number of colours required to paint ${S}^{2}\left(r\right)$ such that any two points of the sphere at unit distance apart have different colours. It is easy to see that the chromatic number of sphere $\chi \left({S}^{2}\left(r\right)\right)$ depends on radius. G. Simmons in 1976 suggested that if $r>1/2$ then $\chi \left({S}^{2}\left(r\right)\right)\ge 4$, and thus $\chi \left({S}^{2}\left(r\right)\right)=4$ for $r\in \left(1/2,{r}^{\ast }\right)$. The main topic of the talk will be the proof of Simmons' conjecture. The proof is technically rather simple and, apart from elementary geometrical constructions, is based on the Borsuk-Ulam theorem. In addition, possible generalisations and some related results will be discussed.

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