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arXiv · 2508.14578

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

Arthur Igorevich Bikeev·Andrei Mikhailovich Raigorodskii
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Abstract

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 $ χ(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 $ χ(n,b) $.

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Arthur Igorevich Bikeev, Andrei Mikhailovich Raigorodskii. 2025-08-20. On upper bounds on the number of parts in the problem of partitioning sets into parts of smaller diameter. https://arxiv.org/abs/2508.14578

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