For integers 1 < k < d-1 and r k+2, we establish new lower bounds on the maximum number of points in nᵈ such that no r lie in a k-dimensional affine (or linear) subspace. These bounds improve on earlier results of Sudakov-Tomon and Lefmann. Further, we provide a randomised construction for the no-four-on-a-circle problem posed by Erdős and Purdy, improving Thiele's bound. We also consider the random construction in higher dimensions, and improve the bound of Suk and White for d 4. In each case, we apply the deletion method, using results from number theory and incidence geometry to solve the associated counting problems.
Ghosal et al. (Mon,) studied this question.