Consider the problem of the chromatic number of a two-dimensional sphere, namely, the smallest number of colours required to paint 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 depends on radius. G. Simmons in 1976 suggested that if then , and thus for . 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.