9 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 circles, and let $p_3({\cal A})$ denote the number of triangular cells. Weak Grünbaum triangle conjec…

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…

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

Strengthened pseudocircle-arrangement conjectures. The following three statements hold:

Infinite-family conjecture. There exists an infinite family of simple arrangement graphs of 4-edge-critical arrangements of pseudocircles.

Obstruction-list conjecture. There is a list-of-obstructions characterization of exactly when a drawing of $G$ has such an extension.

Ortner's conjecture. Embeddability of a strong arrangement into $\Sigma_g$ holds if and only if all its subarrangements consisting of $4+g$ pseudocircles are embeddable into…