Vsevolod Voronov

On the chromatic number of a two-dimensional sphere
 
Consider the problem of the chromatic number of a two-dimensional sphere, namely, the smallest number of colours required to paint $S^2(r)$ 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 $\chi(S^2(r))$ depends on radius.  G. Simmons in 1976 suggested that if $r>1/2$ then $\chi(S^2(r)) \geq 4$, and thus $\chi(S^2(r)) = 4$ for $r \in (1/2, r^*)$.  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.