For the ceiling, he showed the count can never exceed n to the power 1 plus 1 divided by 3, about n^1.333. For the floor, he exhibited explicit point sets where the count reaches at least n to the power 1 plus c divided by log log n, a formula that grows faster than n but only just barely. The chasm between these two bounds — stretching across eight decades and defeating every attempt to close it — is exactly what an AI system cracked open in 2025.
