On upper bounds on the number of parts in the problem of partitioning sets into parts of smaller diameter

In the present paper, we study problems related to the classical Borsuk’s problem. Recall that the Borsuk’s problem consists in finding the smallest number $f(n)$ of parts of smaller diameter into which an arbitrary set of diameter 1 in Euclidean space ${\mathbb{R}}^n$ can be divided. Here we will discuss the quantity $\chi (n,b)$ which differs from the quantity $f(n)$ in that in its definition an arbitrary set of diameter 1 in ${\mathbb{R}}^n$ must be partitioned into parts whose diameters are strictly less than a given number $b\in (0,1]$. In this paper, we collect information about the known upper bounds and, among other things, find a new upper bound for the quantity $\chi (n,b)$.