On Subsets of Lattice Cubes Avoiding Affine and Spherical Degeneracies

For integers $1<k<d-1$ and $r⩾k+2$ , we establish new lower bounds on the maximum number of points in ${\left[n\right]}^d$ 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.