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.
Flammenkamp, Progress in the no-three-in-line probl em, II, Journal of Combinatorial The- ory, Series A 81 (1998), 108–113
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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.