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PreprintCC BY 4.0arXiv

Larger Crossing Families

  • Joachim Orthaber
  • Patrick Schnider
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Abstract

We give the first explicit constant for crossing families of linear size by showing that every set of n points in general position in the plane contains a crossing family of size approximately n/79. We further identify some conditions where there are even larger crossing families. The first such condition forbids the existence of a large different pattern, namely non-convex 4-clusters: We show that every point set contains either a large non-convex 4-cluster or a large crossing family. The second condition is the existence of many crossings: We give a bound for the size of a crossing family in terms of the number of crossings of the complete geometric graph drawn on the point set.

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Cite

BibTeXRIS
BibTeX
@misc{orthaber2026larger,
  title = {{Larger Crossing Families}},
  author = {Joachim Orthaber and Patrick Schnider},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38658},
  eprint = {2610.38658},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38658},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Larger Crossing Families
AU  - Joachim Orthaber
AU  - Patrick Schnider
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38658
UR  - https://arxiv.org/abs/2610.38658
AB  - We give the first explicit constant for crossing families of linear size by showing that every set of n points in general position in the plane contains a crossing family of size approximately n/79. We further identify some conditions where there are even larger crossing families. The first such condition forbids the existence of a large different pattern, namely non-convex 4-clusters: We show that every point set contains either a large non-convex 4-cluster or a large crossing family. The second condition is the existence of many crossings: We give a bound for the size of a crossing family in terms of the number of crossings of the complete geometric graph drawn on the point set.
ER  -