Non-simple Arrangements of Lines and Pseudolines with Many Triangles
Abstract
We study affine arrangements of an even number n of lines and pseudolines with many triangles. For simple ones, Blanc's bound limits the number of triangular faces to ⌊ n(n−5/2)/3⌋. Adapting the Bartholdi-Blanc-Loisel doubling to the even case, we construct an infinite series of arrangements of n=6·2t+2 (8,14,26,50,…) straight lines with two triple points and (n(n−5/2)+1)/3 triangles, one more than the simple bound allows. We obtain exhaustive enumerations of triangle-maximal pseudoline arrangements, simple or not, for even n≤12. For n=6,8,10,12,14 the maximum number of triangles, for both lines and pseudolines, is $7,15,25,38,54$; for n=8,12,14 every optimal arrangement has multiple points. All optimal arrangements for even n≤12 are realizable by straight lines, except two arrangements of 10 pseudolines, which are not stretchable. For the odd value n=11, no arrangement with $33$ triangles exists even with multiple points. We also obtain even arrangements by adding a line to odd ones. For n≡2pmod 6 this again gives one triangle more than the simple bound, in pseudolines up to n=44 and in straight lines for n=20 and n=32. Doubling these pseudoline arrangements gives members of the series 18·2t+2 and 30·2t+2 with the same excess, up to n=146 and n=242, respectively; those with n=38 and n=74 are realizable by straight lines. These results lead us to conjecture that an arrangement of an even number n of pseudolines has at most ⌊ n(n−2)/3⌋−⌊ n/6⌋ triangles.
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BibTeX
@misc{parpalak2026nonsimple,
title = {{Non-simple Arrangements of Lines and Pseudolines with Many Triangles}},
author = {Roman Parpalak and Denis Utkin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03096},
eprint = {2611.03096},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03096},
}
RIS
TY - UNPB
TI - Non-simple Arrangements of Lines and Pseudolines with Many Triangles
AU - Roman Parpalak
AU - Denis Utkin
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03096
UR - https://arxiv.org/abs/2611.03096
AB - We study affine arrangements of an even number n of lines and pseudolines with many triangles. For simple ones, Blanc's bound limits the number of triangular faces to ⌊ n(n−5/2)/3⌋. Adapting the Bartholdi-Blanc-Loisel doubling to the even case, we construct an infinite series of arrangements of n=6·2^t+2 (8,14,26,50,…) straight lines with two triple points and (n(n−5/2)+1)/3 triangles, one more than the simple bound allows. We obtain exhaustive enumerations of triangle-maximal pseudoline arrangements, simple or not, for even n≤12. For n=6,8,10,12,14 the maximum number of triangles, for both lines and pseudolines, is $7,15,25,38,54$; for n=8,12,14 every optimal arrangement has multiple points. All optimal arrangements for even n≤12 are realizable by straight lines, except two arrangements of 10 pseudolines, which are not stretchable. For the odd value n=11, no arrangement with $33$ triangles exists even with multiple points. We also obtain even arrangements by adding a line to odd ones. For n≡2pmod 6 this again gives one triangle more than the simple bound, in pseudolines up to n=44 and in straight lines for n=20 and n=32. Doubling these pseudoline arrangements gives members of the series 18·2^t+2 and 30·2^t+2 with the same excess, up to n=146 and n=242, respectively; those with n=38 and n=74 are realizable by straight lines. These results lead us to conjecture that an arrangement of an even number n of pseudolines has at most ⌊ n(n−2)/3⌋−⌊ n/6⌋ triangles.
ER -