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PreprintarXiv licencearXiv

k-arrangements of pseudolines and pseudocircles

  • Jan Kynčl
  • Carolina Medina
  • Gelasio Salazar
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Abstract

A k-arrangement of pseudolines is a set of bi-infinite curves in the plane such that any two of them intersect each other in exactly k points, at which they cross, and it is simple if no three curves meet at a common point. Cyclic arrangements are the only simple $1$-arrangements of pseudolines that are unavoidable, in the Ramsey spirit: for each fixed m≥ 1, every sufficiently large simple $1$-arrangement of pseudolines has a cyclic subarrangement of size m. We show that, for every m≥ 3, the number of unavoidable simple k-arrangements of pseudolines of size m grows exponentially with k, independently of m. For even k, we prove an analogous result for k-arrangements of pseudocircles.

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Cite

BibTeXRIS
BibTeX
@misc{kyncl2026karrangements,
  title = {{$k$-arrangements of pseudolines and pseudocircles}},
  author = {Jan Kynčl and Carolina Medina and Gelasio Salazar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00773},
  eprint = {2610.00773},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00773},
}
RIS
TY  - UNPB
TI  - k-arrangements of pseudolines and pseudocircles
AU  - Jan Kynčl
AU  - Carolina Medina
AU  - Gelasio Salazar
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00773
UR  - https://arxiv.org/abs/2610.00773
AB  - A k-arrangement of pseudolines is a set of bi-infinite curves in the plane such that any two of them intersect each other in exactly k points, at which they cross, and it is simple if no three curves meet at a common point. Cyclic arrangements are the only simple $1$-arrangements of pseudolines that are unavoidable, in the Ramsey spirit: for each fixed m≥ 1, every sufficiently large simple $1$-arrangement of pseudolines has a cyclic subarrangement of size m. We show that, for every m≥ 3, the number of unavoidable simple k-arrangements of pseudolines of size m grows exponentially with k, independently of m. For even k, we prove an analogous result for k-arrangements of pseudocircles.
ER  -