11 problems

Let $\mathcal C$ be a family of unit disks forming an $m$-fold covering of the plane, meaning that every point belongs to at least $m$ members of $\mathcal C$. Winkler's conjecture…

Let $\mathcal F$ be the family of bottomless rectangles in the plane, and let $m_k^*(\mathcal F)$ denote the corresponding threshold for decomposing coverings into $k$ coverings. B…

Let $\mathcal C$ be a sufficiently thick covering of the plane by unit disks or by smooth convex objects, meaning a covering with enough multiplicity that its members might be deco…

For every positive integer $k$, let $f(k)$ be an integer such that every complete multidigraph whose arcs are the union of $k$ quasi-orders has a dominating set of size at most…

The $S_1$ extremal conjecture.

Let $D\subset\mathbb R^3$ be a convex set, and call a scaled and translated copy of $D$ without rotation a homothetic copy of $D$. Let $S$ be a finite set of points in…

Let $\mathcal A$ be a pseudo-disk arrangement, and let its members be the objects to be colored. The dual pseudo-disk arrangement conjecture. The members of any pseudo-disk arrange…

Let $\mathcal A$ be a pseudo-disk arrangement, meaning a family of planar bodies whose boundaries are Jordan curves and such that the boundary of any member intersects the boundary…

Let $D$ be a convex set in the plane, and call a scaled and translated copy of $D$ without rotation a homothetic copy of $D$. Let $S$ be a finite set of points. The three-color hom…

Let $D$ be a convex set in the plane, let $S$ be a finite set of points, and let $m$ be a positive integer. Pach's conjecture. For every convex set $D$ there is an $m$ such that an…

Let $\mathcal F$ be a family of pseudo-disks in the plane, let $S$ be a finite set of points, and let $m$ be a positive integer. The three-color pseudo-disk conjecture. If…