20 problems

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 ${\cal A}$ be a simple digon-free arrangement of $n$ pairwise intersecting pseudocircles, and let $p_3({\cal A})$ denote the number of triangular cells. Grünbaum's triangle con…

Simplified presentation conjecture. The presentation has the following properties:

Let $\mathcal{A}$ be an arrangement of lines in $\mathbb{CP}^2$, with Milnor fiber $F_\mathcal{A}$ and boundary three-manifold $\partial F_\mathcal{A}$. The combinatorics of…

Maximal-face bound conjecture. This upper bound for $\mu({\mathcal{A}})$ is tight up to a constant factor.

Felsner–Scheucher's conjecture. Every intersecting arrangement of $n\geqslant 3$ pseudocircles has at least $n-1$ triangles.

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.

Erdős et al.'s conjecture. The maximum middle-level complexity satisfies

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 $\widetilde{\mathcal A}$ be the conified arrangement associated with an affine arrangement $\mathcal A$, and let $\Gamma$ be the graph whose vertices are the lines of…

Double-point graph conjecture. If $\Gamma$ is connected, then $\mathcal{A}$ is a-monodromic over $\mathbb Z$.

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 $\Gamma_1$ be a collection of $n$ semi-infinite Jordan curves in the plane, where any two curves intersect at most $s$ times, for some fixed constant $s>0$. Let $A(\Gamma_1)$ b…

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 $F$ be the Milnor fibre associated with a line arrangement, and let $\partial F$ be its boundary. Arrangement recoverability conjecture. The combinatorics of the arrangement ca…

Hosted-triples growth conjecture. For every $\epsilon>0$,

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…

Homeomorphism-type bound conjecture. The number of homeomorphism types occurring in such a parametrized family admits a singly exponential bound in the relevant parameters, in part…