Euclidean Borsuk partition problem in dimensions 4 through 63

Determine whether every bounded subset of Euclidean space of dimension d can be partitioned into d+1 subsets of strictly smaller Euclidean diameter for each dimension 4≤d≤63.

Background

The paper introduces the discrete Borsuk partition problem by analogy with Borsuk’s classical Euclidean partition problem. The Euclidean statement is known to hold in dimensions two and three, while counterexamples are known in sufficiently high dimensions; the intermediate range specified by the authors remains unresolved. This classical problem serves as motivation rather than as a problem solved by the paper.