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

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 $\chi \left(S^2\right(r\left)\right)$ depends on radius. G. Simmons in 1976 suggested that if $r>1/2$ then $\chi \left(S^2\right(r\left)\right)\ge 4$, and thus $\chi \left(S^2\right(r\left)\right)=4$ for $r\in (1/2,r^{\ast })$. 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.