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A note on the no-$(d+2)$-on-a-sphere problem
Published 3 Dec 2024 in math.CO and cs.DM | (2412.02866v1)
Abstract: For fixed $d\geq 3$, we construct subsets of the $d$-dimensional lattice cube $[n]d$ of size $n{\frac{3}{d + 1} - o(1)}$ with no $d+2$ points on a sphere or a hyperplane. This improves the previously best known bound of $\Omega(n{\frac{1}{d-1}})$ due to Thiele from 1995.
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