Set partitions

In this work we consider one important variant of Borsuk’s classical problem of partitioning sets into parts of smaller diameter. It concerns the estimation of the number х(n, b) equal to the minimum number of colors needed to color an arbitrary set of unit diameter in n-dimensional Euclidean space in such a way that the distance between points of the same color is strictly less than b. Lower bounds for х(n, b(n)) for different behavior of b(n) are given.