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arXiv · 2510.17743

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

Alexandr Grebennikov·Matthew Kwan
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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 $n \geqslant k \geqslant 10^{37}$ the maximum number of points is exactly $kn$. Our proof builds on the recent work of Kovács, Nagy, and Szabó (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.

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Alexandr Grebennikov, Matthew Kwan. 2026-09-14. No-$(k+1)$-in-line problem for large constant $k$. https://arxiv.org/abs/2510.17743

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