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

TL;DR

This paper constructs an infinite series of line arrangements with maximum bounded triangles for specific line counts, using computer-assisted verification and exploring potential extensions.

Contribution

It introduces a new infinite series of arrangements achieving maximum bounded triangles for lines of the form 18*2^t+1, verified through computational methods.

Findings

Constructed arrangements for n=18*2^t+1 lines with maximum bounded triangles.

Computer-assisted checks verify conditions for the arrangements.

Evidence suggests no further base configurations exist for certain line counts.

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.