Lax Lean Archive

Large Finite Point Sets Have 4 Collinear Points or a 6-Clique

lax-56·formalized by Édouard Bonnet @EdouardBonnet·registered·created ·GitHub @e351acd·Lean v4.33.0 epoch · mathlib db584cd6d46c

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    Abstract

    We formalize the theorem that every finite set of at least 10245010^{2^{450}} points in the real plane contains either four collinear points or six points that are pairwise visible with respect to the whole set. Visibility means that the open segment joining a pair contains no point of the ambient finite set.

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    @misc{lax-56,
  author = {Édouard Bonnet},
  title = {Large Finite Point Sets Have 4 Collinear Points or a 6-Clique},
  year = {2026},
  howpublished = {Lax Archive, lax-56},
  url = {https://laxarchive.org/lax-56/},
}

    References

    1. Pavel Valtr. On Empty Hexagons. In Surveys on Discrete and Computational Geometry: Twenty Years Later 453:433–441, 2008.
    2. Édouard Bonnet. Large Finite Point Sets Have 4 Collinear Points or a 6-Clique. 2026.
    3. Bálint Hujter and Sándor Kisfaludi-Bak. 5 Colorable Visibility Graphs Have Bounded Size or 4 Collinear Points. 2014. arXiv:1410.7273
    4. Kamil Popielarz, Julian Sahasrabudhe and Richard Snyder. A Stability Theorem for Maximal K_r+1-free Graphs. Journal of Combinatorial Theory, Series B 132:236–257, 2018.

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