Arrangements of pseudohyperplanes are widely studied in computational geometry. A rich subclass of pseudohyerplane arrangements, which has gained more attention in recent years, is the so-called signotopes. Introduced by Manin and Schechtman (1989), the higher Bruhat order is a natural order of r-signotopes on n elements, with the signotope corresponding to the cyclic arrangement as the minimal element. In this paper, we show that the lower (and by symmetry upper) levels of this higher Bruhat order contain the same number of elements for a fixed difference n-r. This result implies that given the difference d = n-r and p, the number of one-element extensions of the cyclic arrangement of n hyperplanes in ℝ^d with at most p points on one side of the extending pseudohyperplane does not depend on n, as long as n ≥ d + p.

A conjecture by Rafla from 1988 asserts that every simple drawing of the complete graph $K_n$ admits a plane Hamiltonian cycle. It turned out that already the existence of much simpler non-crossing substructures in such drawings is hard to prove. Recent progress was made by Aichholzer et al. and by Suk and Zeng who proved the existence of a plane path of length $Ω(\log n / \log \log n)$ and of a plane matching of size $Ω(n^{1/2})$ in every simple drawing of $K_n$. Instead of studying simpler substructures, we prove Rafla's conjecture for the subclass of convex drawings, the most general class in the convexity hierarchy introduced by Arroyo et al. Moreover, we show that every convex drawing of $K_n$ contains a plane Hamiltonian path between each pair of vertices (Hamiltonian connectivity) and a plane $k$-cycle for each $3 \leq k \leq n$ (pancyclicity), and present further results on maximal plane subdrawings.

Gons and holes in point sets have been extensively studied in the literature. For simple drawings of the complete graph a generalization of the Erdős--Szekeres theorem is known and empty triangles have been investigated. We introduce a notion of $k$-holes for simple drawings and survey generalizations thereof, like empty $k$-cycles. We present a family of simple drawings without $4$-holes and prove a generalization of Gerken's empty hexagon theorem for convex drawings. A crucial intermediate step is the structural investigation of pseudolinear subdrawings in convex drawings. With respect to empty $k$-cycles, we show the existence of empty $4$-cycles in every simple drawing of $K_n$ and give a construction that admits only $Θ(n^2)$ of them.

In 1926, Levi showed that, for every pseudoline arrangement $\mathcal{A}$ and two points in the plane, $\mathcal{A}$ can be extended by a pseudoline which contains the two prescribed points. Later extendability was studied for arrangements of pseudohyperplanes in higher dimensions. While the extendability of an arrangement of proper hyperplanes in $\mathbb{R}^d$ with a hyperplane containing $d$ prescribed points is trivial, Richter-Gebert found an arrangement of pseudoplanes in $\mathbb{R}^3$ which cannot be extended with a pseudoplane containing two particular prescribed points. In this article, we investigate the extendability of signotopes, which are a combinatorial structure encoding a rich subclass of pseudohyperplane arrangements. Our main result is that signotopes of odd rank are extendable in the sense that for two prescribed crossing points we can add an element containing them. Moreover, we conjecture that in all even ranks $r \geq 4$ there exist signotopes which are not extendable for two prescribed points. Our conjecture is supported by examples in ranks 4, 6, 8, 10, and 12 that were found with a SAT based approach.