9 problems

Simplified presentation conjecture. The presentation has the following properties:

Grünbaum's digon conjecture. Every simple arrangement of $n>2$ pairwise crossing pseudocircles has at most $2n-2$ digons.

Let ${\cal A}$ be a simple arrangement of $n\ge 3$ pairwise intersecting pseudocircles, and let $p_3({\cal A})$ denote the number of triangular cells. Triangle lower-bound conjectu…

Let $\Delta(j,k)$ be the minimal dimension $d$ of a Euclidean space $\mathbb{R}^d$ such that every collection of $j$ masses in $\mathbb{R}^d$ admits a $k$-arrangement equiparting a…

Felsner–Hurtado–Noy–Streinu conjecture. The arrangement graph of every simple arrangement of great-circles on the sphere is 3-colorable.

Let $d\geq 3$ and $n\geq 1$ be integers, and let $S^{d-1}$ be covered by the union of $n$ congruent spherical zones. Suppose each point of $S^2$ belongs to the interior of at most…

Let $C_n$ be the set of numbers of regions formed by $n$ circles in the plane, in an arrangement not in general position, such that every two circles have at least one common point…

Let ${\mathcal A}$ be a decomposable arrangement, and let $G({\mathcal A})$ denote its arrangement group. Embedding conjecture. The group $G({\mathcal A})$ embeds in a product of f…

Let $\mathcal{A}$ be an affine arrangement or a projective arrangement of $n$ pseudo-lines. The quantities $a_3(\mathcal{A})$ and $p_3(\mathcal{A})$ denote the numbers of triangula…