Liam Kruer

math.CO — Combinatorics

Shared triangular sides in pseudoline arrangements

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

Let $d$ 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 $t_3$ is the number of triple crossings, we prove $d\leq 3t_3$, 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.