Leírás
For every $d\geq 2$, we construct a subset $\Gamma\subseteq \{1,\dots,n\}^d$ of size $n-o(n)$ in general position (i.e. every affine hyperplane of $\mathbb{R}^d$ intersects $\Gamma$ in at most $d$ points) such that every hypersphere of $\mathbb{R}^n$ intersects $\Gamma$ in at most $d+1$ points. This construction is the largest one currently known for the $d$-dimensional general position no-$(d+2)$-on-a-sphere problem (originating from Erdős and Purdy), and improves Thiele's $\frac{3}{32}n^{1/(d-1)}$, and the recent $\frac{1}{d+1}n-o(n)$ bound of Dong and Xu.
More generally, we prove that the role of hyperspheres above can be replaced by $Q$-quadrics, i.e. by quadratic surfaces given by a polynomial whose degree two homogeneous part equals a fixed quadratic form $Q$. Furthermore, we formulate analogous statements in affine spaces over (finite) fields.
These constructions are all based on a suitable rational normal curve in a $d$-dimensional projective space obtained by generalising the ideas of Dong, Xu, and Thiele.
Available at https://arxiv.org/abs/2511.03526.