The Kobon triangle problem on triangles in line arrangements

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

An arrangement consists of $n$ distinct straight lines in $\mathbb R^2$. The lines subdivide the complement into regions. A triangular region is a bounded region whose closure has three sides, each lying on a line of the arrangement; its interior is crossed by no arrangement line. Distinct such regions have disjoint interiors. Parallel lines and multiple concurrent lines are allowed; no restriction to general position is imposed. Let $T(n)$ be the largest possible number of triangular regions over all such arrangements.

Statement recorded in the supplied source.

Determine $T(n)$, the maximum number of nonoverlapping triangles that can be formed by an arrangement of $n$ straight lines, for each $n\geq1$. [S2]

Short English statement

How many triangular cells can an arrangement of a prescribed number of straight lines produce?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200050 (AMR-071-0050). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Pavlo Savchuk, Constructing Optimal Kobon Triangle Arrangements via Table Encoding, SAT Solving, and Heuristic Straightening (2026 version of 2025 preprint). Sections 1 and 2.1–2.2, pp. 1–3. https://arxiv.org/pdf/2507.07951.