Randomised algebraic constructions for the no-$(k+1)$-in-line problem

Abstract

The no-(k+1)-in line problem seeks the maximum number of points that can be selected from an $n\times n$ square lattice such that no $k+1$ of them are collinear. The problem was first posed more than $100$ years ago for the special case $k=2$ and has remained open ever since. The general problem was recently resolved in the case $k$ is not small compared to $n$, as Kovács, Nagy and Szabó proved that the upper bound $kn$ can be attained, provided that $k>C\sqrt{n\log n}$ for an absolute constant $C$. In this paper, we show that $\left(1-\frac{2}{k}\right)kn\le f_k(n)\le kn$ and $\left(1-\frac{3}{k}\right)kn\le f_k(n)\le kn$ hold for every even $k$ and odd $k$, respectively, provided that $n$ is large enough. This is asymptotically tight as $k\to \infty$. Previously, only $f_k(n)=\hat{I}\text{c◯}(kn)$ was known due to Lefmann. We present further improvements on the lower bounds for constant values of $k$ when $k<23$ holds. All these bounds are based on randomised algebraic constructions.

28 pages+Appendix, 17 figures. To be published on 18 September 2026