The $18\cdot 2^t+1$ Triangle-Maximal Series of Straight Lines
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915952886546432 |
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| author | Parpalak, Roman Utkin, Denis |
| author_facet | Parpalak, Roman Utkin, Denis |
| contents | Given $n$ lines in general position in the plane, how many bounded triangular faces can the arrangement have? We construct a straight-line affine arrangement of $19$ lines satisfying the conditions of the iterative construction by Bartholdi, Blanc, and Loisel, thereby obtaining an infinite series of straight-line arrangements attaining the maximum number of bounded triangles for every $n=18\cdot 2^t+1$. The conditions are verified by computer-assisted interval and combinatorial checks. A computational search over $n=21$, $23$, $27$ lines provides strong evidence against the existence of further base configurations compatible with the known iterative constructions, but reveals arrangements allowing a single iterative step that yield arrangements of $41$ and $45$ lines with $533$ and $645$ bounded triangles, respectively, each matching the upper bound. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_22035 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $18\cdot 2^t+1$ Triangle-Maximal Series of Straight Lines Parpalak, Roman Utkin, Denis Combinatorics 52C30 (Primary) 52C35, 05B30 (Secondary) Given $n$ lines in general position in the plane, how many bounded triangular faces can the arrangement have? We construct a straight-line affine arrangement of $19$ lines satisfying the conditions of the iterative construction by Bartholdi, Blanc, and Loisel, thereby obtaining an infinite series of straight-line arrangements attaining the maximum number of bounded triangles for every $n=18\cdot 2^t+1$. The conditions are verified by computer-assisted interval and combinatorial checks. A computational search over $n=21$, $23$, $27$ lines provides strong evidence against the existence of further base configurations compatible with the known iterative constructions, but reveals arrangements allowing a single iterative step that yield arrangements of $41$ and $45$ lines with $533$ and $645$ bounded triangles, respectively, each matching the upper bound. |
| title | The $18\cdot 2^t+1$ Triangle-Maximal Series of Straight Lines |
| topic | Combinatorics 52C30 (Primary) 52C35, 05B30 (Secondary) |
| url | https://arxiv.org/abs/2604.22035 |