A new construction gives subsets of the d-dimensional lattice cube of size n^{3/(d+1)-o(1)} with no d+2 points on a sphere or hyperplane, improving Thiele's 1995 bound.
Dudeney, Amusements in Mathematics , Nelson, London, 1917
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A note on the no-$(d+2)$-on-a-sphere problem
A new construction gives subsets of the d-dimensional lattice cube of size n^{3/(d+1)-o(1)} with no d+2 points on a sphere or hyperplane, improving Thiele's 1995 bound.