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Shared triangular sides in pseudoline arrangements

Primarily AI-generated textHuman understanding: no partsmath.CO — Combinatorics

Contributed by Will Howe, Grant Huffman ↗, Jensen Kohlmeyer ↗, Liam Kruer ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

Let dd be the number of edges incident to two bounded triangular faces in a finite affine arrangement of pseudolines, with each pair crossing exactly once and with only double and triple crossings. If t3t_3 is the number of triple crossings, we prove d≤3t3d\leq 3t_3, answering the compensation question of Parpalak and Utkin. The proof distributes charge along the normalized boundary of the union of triangular faces. A double crossing incident to four triangles is treated by restricting the arrangement to the four pseudolines supporting their opposite sides. This restriction provides two boundary edges whose unused endpoint charges pay for the crossing.

Provenance statement

OpenAI Codex assisted with mathematical exploration, development of the argument, and preparation of the manuscript, as acknowledged in the paper. The authors report that the text is primarily AI-generated and that none of the contribution has been checked and understood by a human. For this submission, Codex transcribed the supplied November 8, 2026 PDF into TeX and preserved its original vector figure. A companion Lean formalization was shared with the manuscript.

Tools used

OpenAI
Codex

References

  1. R. Parpalak and D. Utkin, Non-simple arrangements of lines and pseudolines with many triangles , arXiv:2611.03096v1 (2026), https://arxiv.org/abs/2611.03096.arXiv

Version history

  1. v1 · 4 pagesSubmitted by Liam KruerInitial depositCurrentNov 09, 2026