14 problems

Let $k$ be a fixed positive integer, and let a family consist of $n$ pairwise intersecting bi-infinite $x$-monotone curves such that any two curves intersect at most $k$ times. Lin…

A $k$-quasiplanar graph is a graph that can be drawn in the plane with no $k$ pairwise crossing edges. The linear edge bound conjecture. For any $k>1$, a $k$-quasiplanar graph on…

A complete $n$-vertex simple topological graph is a drawing of the complete graph on $n$ vertices in which vertices are points and edges are simple curves, with no edge passing thr…

Let $h(n)$ denote the maximum, over complete $n$-vertex simple topological graphs, of the minimum number of crossings of an edge in a suitable noncrossing matching construction. Sh…

Let $\mathcal{P}$ be a set of $n$ points in the plane, and let there be $m$ topological trees such that every pair of trees intersects exactly once and every leaf of every tree bel…

Crossing-degree conjecture. There is an absolute constant $\varepsilon>0$ such that

Plane-path conjecture. There is an absolute constant $\varepsilon>0$ such that every complete $n$-vertex simple topological graph contains a plane path of length $n^{\varepsilon}$.

Eulerian edge-refinement conjecture. $d$-spheres have edge refinements which are Eulerian.

Eulerian edge-refinement conjecture. It is possible to edge refine $H$, using only edges different from those of $G$, so that the modified $H$ becomes Eulerian.

Sphere coloring conjecture. Every $d$-sphere has chromatic number $d+1$ or $d+2$.

Call a topological graph $k$-quasi-plane if it has no $k$ pairwise crossing edges. The quasi-plane graph linear density conjecture. For any integer $k \geq 2$ there is a constant…

Bounded-intersection hub curve conjecture. For every $t$, the intersection graph of $\mathcal{F}$ is $\chi$-bounded.

Angularly monotone matching conjecture. Every simple angularly monotone complete topological graph on $n$ vertices contains a disjoint matching of size

A topological graph is a graph drawn in the plane with vertices represented by points and edges by curves connecting their endpoints; two edges cross when their curves intersect aw…