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PreprintarXiv licencearXiv

Large Planar Point Sets Contain 4 Collinear Points or Almost 7-Cliques, and Related Results

  • Bhaswar B. Bhattacharya
  • Sandip Das
  • Sk Samim Islam
  • Aashirwad Mohapatra
  • Saumya Sen
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Abstract

We prove that every sufficiently large finite planar point set contains either four collinear points or seven points with at most one non-visible pair. More generally, we show that for every fixed graph H with chromatic number at most five, or with chromatic number six and a color-critical edge, the visibility graph of every sufficiently large finite planar point set with no four collinear points contains a copy of H. These results extend the recent breakthrough of Bonnet (2026), guaranteeing six pairwise visible points, and come within one visibility edge of the next open case of the big-line-big-clique conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{bhattacharya2026large,
  title = {{Large Planar Point Sets Contain 4 Collinear Points or Almost 7-Cliques, and Related Results}},
  author = {Bhaswar B. Bhattacharya and Sandip Das and Sk Samim Islam and Aashirwad Mohapatra and Saumya Sen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25727},
  eprint = {2609.25727},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25727},
}
RIS
TY  - UNPB
TI  - Large Planar Point Sets Contain 4 Collinear Points or Almost 7-Cliques, and Related Results
AU  - Bhaswar B. Bhattacharya
AU  - Sandip Das
AU  - Sk Samim Islam
AU  - Aashirwad Mohapatra
AU  - Saumya Sen
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.25727
UR  - https://arxiv.org/abs/2609.25727
AB  - We prove that every sufficiently large finite planar point set contains either four collinear points or seven points with at most one non-visible pair. More generally, we show that for every fixed graph H with chromatic number at most five, or with chromatic number six and a color-critical edge, the visibility graph of every sufficiently large finite planar point set with no four collinear points contains a copy of H. These results extend the recent breakthrough of Bonnet (2026), guaranteeing six pairwise visible points, and come within one visibility edge of the next open case of the big-line-big-clique conjecture.
ER  -