Classical Borsuk partition problem in dimensions four through sixty-three

Determine whether every bounded subset of R^d can be partitioned into d+1 subsets of strictly smaller Euclidean diameter for dimensions 4≤d≤63.

Background

The introduction reviews the classical Euclidean Borsuk partition problem, which asks whether every bounded subset of R^d can be partitioned into d+1 parts of smaller diameter. Positive results are known in dimensions two and three, while counterexamples exist in dimensions 64 and higher. The unresolved range explicitly identified by the authors is dimensions four through sixty-three.