Eventual classification of simplicial line arrangements
A finite set of distinct lines in $\mathbb{RP}^2$ is a simplicial arrangement if every connected component of its complement is a triangle. Two arrangements are combinatorially equivalent if their induced incidence structures of vertices, edges and faces are isomorphic. Grünbaum's catalogue of simplicial arrangements consists of the three infinite families listed below and about one hundred sporadic arrangements, the largest with $37$ lines, and Grünbaum conjectured that its list of sporadic arrangements is complete. Cuntz found four further sporadic arrangements in a computer enumeration of simplicial arrangements of at most $27$ lines, and later one of $35$ lines. Panov and Tahar proved, using the structure theorem of Green and Tao for arrangements with few double points, that for every $K > 0$ only finitely many combinatorial classes of simplicial arrangements of $n$ lines with at most $Kn$ double points lie outside the three families.
Problem. Is there an integer $N$ such that every simplicial arrangement of at least $N$ lines is combinatorially equivalent to one of the following?
- A pencil of $k \ge 2$ lines through a common point, together with one line not through that point.
- The $k$ lines supporting the sides and the $k$ symmetry axes of a regular $k$-gon, $k \ge 3$.
- The arrangement in 2 for an even $k \ge 4$, together with the line at infinity.
No list of the arrangements with fewer than $N$ lines is asked for.
Source: Grünbaum (1971), conjectured combinatorial classification of simplicial arrangements (catalogue 2009). Stated in this asymptotic form by AIM — Open Applied Problems, after Panov and Tahar (2026), introduction, who call it Grünbaum's conjectural asymptotic classification.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CO — Combinatorics
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, line arrangements, simplicial arrangements
References
@inproceedings{grunbaum1971,
author = {Gr{\"u}nbaum, Branko},
title = {Arrangements of hyperplanes},
booktitle = {Proceedings of the Second Louisiana Conference on Combinatorics, Graph Theory and Computing},
publisher = {Louisiana State University},
address = {Baton Rouge, LA},
year = {1971},
pages = {41--106},
}
@article{grunbaum2009,
author = {Gr{\"u}nbaum, Branko},
title = {A catalogue of simplicial arrangements in the real projective plane},
journal = {Ars Math. Contemp.},
volume = {2},
number = {1},
year = {2009},
pages = {1--25},
doi = {10.26493/1855-3974.88.e12},
}
@article{panov2026,
author = {Panov, Dmitri and Tahar, Guillaume},
title = {Simplicial arrangements with few double points},
journal = {Discrete Comput. Geom.},
volume = {76},
number = {2},
year = {2026},
pages = {958--977},
doi = {10.1007/s00454-025-00798-3},
eprint = {2409.01892},
archivePrefix = {arXiv},
}
@article{cuntz2012,
author = {Cuntz, Michael},
title = {Simplicial arrangements with up to 27 lines},
journal = {Discrete Comput. Geom.},
volume = {48},
number = {3},
year = {2012},
pages = {682--701},
doi = {10.1007/s00454-012-9423-7},
eprint = {1108.3000},
archivePrefix = {arXiv},
}
@article{cuntz2022,
author = {Cuntz, Michael},
title = {A greedy algorithm to compute arrangements of lines in the projective plane},
journal = {Discrete Comput. Geom.},
volume = {68},
number = {1},
year = {2022},
pages = {107--124},
doi = {10.1007/s00454-021-00351-y},
eprint = {2006.14431},
archivePrefix = {arXiv},
}
@article{green2013,
author = {Green, Ben and Tao, Terence},
title = {On sets defining few ordinary lines},
journal = {Discrete Comput. Geom.},
volume = {50},
number = {2},
year = {2013},
pages = {409--468},
doi = {10.1007/s00454-013-9518-9},
eprint = {1208.4714},
archivePrefix = {arXiv},
}
@misc{aim-simplicial-arrangement-classification,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Eventual classification of simplicial line arrangements}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/625-simplicial-arrangement-classification.md},
note = {Entry 625 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00899, "Eventual classification of simplicial line arrangements", clopen.solutions.