When I formulate the Borsuk conjecture in the form of a coloring problem, Aigner told me that the formulation is not quite correct: It only works if the set is finite. He's right.
All the constructions for counter-examples to Borsuk conjecture use finite set (thanks to Larman's conjecture?). It seems to me that we might obtain infinite counter-examples from them by, for example, taking convex hulls. Am I right?
So what do we known about the existence of infinite counter-examples from the existence of finite counter-examples, and vice versa? Can we somehow use de Brujin–Erdős Theorem？ Is it possible that, in some dimensions, only infinite counter-examples exist?
According to Wikipedia, Borsuk conjecture is known to be true for smooth convex bodies, centrally-symmetric bodies and evolution bodies.
