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) $.
Categories: math.CO