4 problems

Existence conjecture. Completely saturated packings and completely reduced coverings exist for every body $K$ in either $n$-dimensional Euclidean or hyperbolic space.

Let $G$ be a graph and let $A$ be a subset of the vertices of $G$. An $A$-path is a path in $G$ whose two endpoints belong to $A$. For subsets $X$ of vertices of $G$ and integers…

Fejes Tóth's hexagonal density conjecture. For any $\mu\in(\sqrt{3}-1,1)$ and any $\mu$-arrangement $\mathcal{F}$ in $\mathbb{R}^2$,

Let $P$ be the relevant parameter region containing $(1.1)$, let $\Omega$ be the set of packing and covering density pairs for convex disks, and let the rays be $\{(x,1):x<1\}$ and…