Lax Lean Archive

Draft — mutable and not usable as a dependency; its citation marks the draft state.

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

lax-56·formalized by Édouard Bonnet @EdouardBonnet·created 2026-08-20·GitHub @813f264·Lean v4.30.0 epoch · mathlib c5ea00351c28

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this submission may be incorrect.

No flags have been submitted.

    Community review

    Flag this submission

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    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.

    Concepts

    thm✓proven claimdefdefinition

    Concept map

    Proven claimDefinitionThis submissionA → B: B builds on A

    Proofs

    Proof networkview on GitHub

    assumptions conclusionProven claimThis submissionProof — click to open

    Lean sources for these proofs: proofs/ on GitHub

    Proof code is not displayed; the archive records each proof's checked relationship between claims.

    Related submissions

    No other submission in the archive builds on this one, and this one builds on none.

    Cite this

    @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/},
  note = {draft},
}

    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.

    Community review

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above; your ORCID profile must share a public name.

    0 comments

    Loading discussion…