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arXiv · 2604.22035

The $18\cdot 2^t+1$ Triangle-Maximal Series of Straight Lines

Roman Parpalak·Denis Utkin
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Abstract

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.

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Roman Parpalak, Denis Utkin. 2026-04-23. The $18\cdot 2^t+1$ Triangle-Maximal Series of Straight Lines. https://arxiv.org/abs/2604.22035

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