On Triangles in Colored Pseudoline Arrangements

We consider the faces in pseudoline arrangements in which the pseudolines are colored with two colors. Björner, Las Vergnas, Sturmfels, White, and Ziegler conjecture the existence of a two-colored triangle in such arrangements. We consider variants of this problem. We show that in any non-trivial two-coloring of a pseudoline arrangement there exists a two-colored triangle or quadrangle. We also investigate the existence of a bichromatic triangle assuming certain structures on the coloring. Previously, several authors investigated the chromatic number and independence number of hypergraphs whose vertices correspond to the pseudolines of an arrangement and the hyperedges correspond to the faces of the arrangement. We show that the maximum of the independence numbers of such hypergraphs is $\lceil \frac{2}{3}n-1\rceil$. We also prove that if we only consider the triangular faces then this maximum becomes $n-Θ(\log n)$.

💡 Research Summary

The paper studies faces in simple Euclidean pseudoline arrangements whose pseudolines are colored with two colors (red and blue). The original conjecture of Björner, Las Vergnas, Sturmfels, White and Ziegler (1993) asserts that every such bicolored arrangement contains a bichromatic triangle, i.e., a bounded triangular face supported by two pseudolines of one color and one of the other. While the conjecture remains open for general arrangements, the authors obtain several partial results, introduce new structural tools, and analyze related hypergraphs.

Main combinatorial results.

1. Theorem 2 (≤5 red lines). If a bicolored arrangement has at most five red pseudolines (and at least one blue line), then it necessarily contains a bichromatic triangle. The proof proceeds by induction on the number of red lines. For ≤2 red lines the claim is trivial because each line participates in a triangle and the triangle cannot be monochromatic. For three to five red lines the authors use the notion of an extremal pseudoline (a line whose crossings all lie on one side). If an extremal red line is also extremal in the whole arrangement it can be removed and the induction hypothesis applied. If not, Lemma 1 (the sweeping lemma) guarantees a triangle supported by that line on the side without red‑red crossings, which must be bichromatic. The only configuration without an extremal red line for five lines is the “5‑star”. A careful analysis of the 5‑star together with a second application of Lemma 1 shows that a bichromatic triangle must appear, possibly after a single triangle flip.

2. Theorem 3 (block‑bicolored arrangements). An arrangement is called block‑bicolored if, after fixing a north face, the first k pseudolines in the induced ordering are red and the remaining n−k are blue. Using the language of signotopes (rank‑3 orientation functions) the authors show that any block‑bicolored arrangement lies between the all‑negative and all‑positive signotopes in the inclusion order. By Theorem 1 (Felsner–Weil) the inclusion order coincides with the single‑step inclusion order, so there exists a sequence of “−→+” triangle flips transforming the arrangement into the all‑positive signotope. Because the monotone (single‑color) triples are unchanged, each flip involves a bichromatic triangle, and the first flip already yields a bichromatic triangle in the original arrangement.

3. Theorem 4 (triangle or quadrangle). For an arbitrary bicolored arrangement the authors prove a weaker but unconditional statement: either a bichromatic triangle exists, or there is a bichromatic quadrangle whose four supporting pseudolines have the color pattern red‑red‑blue‑blue. The proof combines the previous two theorems and a case analysis of the possible minimal face configurations when no bichromatic triangle is present.

These three theorems together settle the conjecture for all arrangements with at most eleven pseudolines (since any arrangement with ≤11 lines has ≤5 red lines after a suitable recoloring) and for the whole class of block‑bicolored arrangements, which includes many natural families (e.g., arrangements obtained by shifting a block of lines).

Hypergraph perspective.
The authors consider two families of 3‑uniform hypergraphs derived from an arrangement A of n pseudolines:

• Line‑face hypergraph H_{psline‑face}(A): vertices are the pseudolines; each (bounded or unbounded) face yields a hyperedge consisting of the pseudolines that support that face.

• Line‑triangle hypergraph H_{psline‑triangle}(A): the same vertex set, but hyperedges correspond only to triangular faces (including the five unbounded triangular regions).

Previous work (Bose et al., Ackerman et al., Balogh–Solymosi) gave bounds on the independence number α and chromatic number χ of these hypergraphs, but the exact maximum of α over all arrangements was unknown.

4. Theorem 6 (max independence for line‑face). The authors prove that for any n,
    max_{|A|=n} α(H_{psline‑face}(A)) = ⌈2n/3 – 1⌉.
   For line arrangements (as opposed to pseudolines) they obtain the slightly weaker bound
    ⌊2(n−1)/3⌋ ≤ max α ≤ ⌈2n/3 – 1⌉.
   The construction achieving the upper bound uses a periodic pattern of three‑cycles; the lower bound follows from a careful partition of the arrangement into triples and applying the pigeon‑hole principle.

5. Theorem 7 (max independence for line‑triangle). When only triangular faces are considered, the independence number can be much larger:
    max_{|A|=n} α(H_{psline‑triangle}(A)) = n – Θ(log n).
   Thus one can select almost all pseudolines without creating a monochromatic triangle, but a logarithmic number of lines is unavoidable. The proof adapts the construction from the Erdős (3,4) problem (used in