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

Benedek Kovács; Zoltán Lóránt Nagy; Dávid R. Szabó

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

Combinatorics 2025-02-04 v1

Authors: Benedek Kovács , Zoltán Lóránt Nagy , Dávid R. Szabó

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.

Keywords

combinatorial bounds and extremal problems incidence geometry of points and lines collatz conjecture

Cite

@article{arxiv.2502.00176,
  title  = {Settling the no-$(k+1)$-in-line problem when $k$ is not small},
  author = {Benedek Kovács and Zoltán Lóránt Nagy and Dávid R. Szabó},
  journal= {arXiv preprint arXiv:2502.00176},
  year   = {2025}
}

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13 pages, 2 figures

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

Abstract

Keywords

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