Worked solution: Kahn–Kalai's disproof of Borsuk's conjecture via the Frankl–Wilson theorem

Step 7 of 7: The exact threshold dimension

d^\ast

remains open

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In plain words

So Borsuk's conjecture is now known to be both true (in low dimensions and for round or symmetric shapes) and false (in high dimensions, via these combinatorial cuts); the natural next question — exactly where does the switch happen? — turns out to be much harder than either half of that story, and nobody currently knows the precise smallest dimension where a counterexample can exist.

d^\ast \le 63

Detailed analysis

Borsuk's conjecture is classically known to be true for $n \le 3$ and for all smooth convex bodies in any dimension; Kahn and Kalai's disproof only shows failure for large $n$ . Subsequent explicit constructions push the smallest known failing dimension $d^\ast$ down to $n = 64$ (rigorously verified) or conjecturally $n = 63$ (Ji's computer/AI-assisted preprint, not yet peer-reviewed), but the exact value of $d^\ast$ is still unknown.

Knowledge used in this step

Convex and discrete geometry