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PreprintarXiv licencearXiv

The extensible no-four-on-a-circle problem

  • Anubhab Ghosal
  • Ritesh Goenka
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Abstract

We show that there exists a set S ⊂ ℤ2 containing no four points on a circle or a line such that |S ∩ [n]2| = Ω(n) as n → ∞. Since any no-four-on-a-circle set in [n]2 has size O(n), this resolves (up to a constant) a question raised by the current authors and Keevash concerning the density of extensible no-four-on-a-circle constructions. Our construction is based on weighted random sampling from the integer lattice followed by careful deletion.

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Cite

BibTeXRIS
BibTeX
@misc{ghosal2026extensible,
  title = {{The extensible no-four-on-a-circle problem}},
  author = {Anubhab Ghosal and Ritesh Goenka},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20447},
  eprint = {2609.20447},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20447},
}
RIS
TY  - UNPB
TI  - The extensible no-four-on-a-circle problem
AU  - Anubhab Ghosal
AU  - Ritesh Goenka
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20447
UR  - https://arxiv.org/abs/2609.20447
AB  - We show that there exists a set S ⊂ ℤ^2 containing no four points on a circle or a line such that |S ∩ [n]^2| = Ω(n) as n → ∞. Since any no-four-on-a-circle set in [n]^2 has size O(n), this resolves (up to a constant) a question raised by the current authors and Keevash concerning the density of extensible no-four-on-a-circle constructions. Our construction is based on weighted random sampling from the integer lattice followed by careful deletion.
ER  -