Counterexamples to Borsuk's Conjecture on Spheres of Small Radius

Borsuk's classical problem of combinatorial geometry is considered and counterexamples to Borsuk's Conjecture on spheres of small radius are studied. To construct a counterexample to Borsuk's conjecture, the required set containing multidimensional simplices are assumed. Abstract graphs should contain no cliques and should have sufficiently large chromatic number. The results show that counterexamples to Borsuk's conjecture can be constructed on spheres with radii substantially smaller than 0.707. A theorem is proved showing that the Borsuk's conjecture is false for any value greater than 0.5.