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arXiv · 2502.00176

Settling the no-$(k+1)$-in-line problem when $k$ is not small

Benedek Kovács·Zoltán Lóránt Nagy·Dávid R. Szabó
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Abstract

What is the maximum number of points that can be selected from an $n \times n$ square lattice such that no $k+1$ of them are in a line? This has been asked more than $100$ years ago for $k=2$ and it remained wide open ever since. In this paper, we prove the precise answer is $kn$, provided that $k>C\sqrt{n\log{n}}$ for an absolute constant $C$. The proof relies on carefully constructed bi-uniform random bipartite graphs and concentration inequalities.

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Benedek Kovács, Zoltán Lóránt Nagy, Dávid R. Szabó. 2025-01-31. Settling the no-$(k+1)$-in-line problem when $k$ is not small. https://arxiv.org/abs/2502.00176

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