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

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arxiv 2502.00176 v1 pith:LFXMDLKZ submitted 2025-01-31 math.CO

Benedek Kovács , Zoltán Lóránt Nagy , Dávid R. Szabó This is my paper
classification math.CO
keywords absoluteansweraskedbeenbi-uniformbipartitecarefullyconcentration
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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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  1. No-$(k+1)$-in-line problem for $k \geqslant 3$

    math.CO 2026-07 accept novelty 8.0 of 10

    For k≥3 and sufficiently large n, the maximum number of points in an n×n grid with no k+1 collinear is exactly kn.

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