No-$(k+1)$-in-line problem for large constant $k$

TL;DR

This paper determines the maximum number of points in an n-by-n grid so that no line contains more than k points, proving it is exactly kn for very large k, and extends results to higher dimensions.

Contribution

It establishes the exact maximum for large constant k in the grid line problem, improving previous bounds and extending to higher dimensions.

Findings

Maximum points is exactly kn for k ≥ 10^37 and n ≥ k.

Builds on recent work, combining ideas from Kovács, Nagy, Szabó, Jain, and Pham.

Provides new bounds for higher-dimensional versions of the problem.

Abstract

How many points can be placed in an $n\times n$ grid so that every (affine) line contains at most $k$ points? We prove that for $n \ge k \ge 10^{37}$ the maximum number of points is exactly $kn$. Our proof builds on the recent work of Kov\'acs, Nagy, and Szab\'o (who proved an analogous result when $k$ is at least about $\sqrt{n \log n}$), incorporating ideas of Jain and Pham. Using the same approach, we also obtain new bounds for higher-dimensional extensions of this problem.