The no-three-in-line problem on a torus

Abstract

Let $T({\mathbb{Z}}_m\times {\mathbb{Z}}_n)$ denote the maximal number of points that can be placed on an $m\times n$ discrete torus with "no three in a line," meaning no three in a coset of a cyclic subgroup of ${\mathbb{Z}}_m\times {\mathbb{Z}}_n$. By proving upper bounds and providing explicit constructions, for distinct primes $p$ and $q$, we show that $T({\mathbb{Z}}_p\times {\mathbb{Z}}_{p^2})=2p$ and $T({\mathbb{Z}}_p\times {\mathbb{Z}}_{pq})=p+1$. Via Gröbner bases, we compute $T({\mathbb{Z}}_m\times {\mathbb{Z}}_n)$ for $2\le m\le 7$ and $2\le n\le 19$.

10 pages, 3 figures