Progress on Borsuk’s Problem and Answer to TYI 60.

The Finite and the Infinite Borsuk problem TYI(60) The question Let f(d) be the smallest integer so that every set of diameter one in can be covered by f(d) sets of smaller diameter. Let g(d) be the smallest integer so that every finite set of diameter one in can be covered by g(d) sets of smaller diameter. Clearly g(d) ≤ f(d) Test your intuition: Is g(d)=f(d) ? TYI (60): my intuition The answer is not known, but my own intuition is that g(d)<f(d) for every large d. In my lecture around the Borsuk Conjecture at Jeff Kahn’s fest I raised the conjecture that this is indeed the case. I also mentioned an interesting psychological fact: Over the years, both before and after I wrote my 2015 survey article on problems around Borsuk’s conjecture, I have encountered several incorrect proofs that g(d) = f(d). Often these proofs involve the De Bruijn–Erdős theorem, which asserts that an infinite graph with finite chromatic number has a finite subgraph with the same chromatic number. (This result…