16 problems

Let $n\geq 3$ and let $K$ be an $n$-dimensional convex body. A positive homothetic copy of $K$ is a set of the form $x+\lambda K$ with $x\in\mathbb{R}^n$ and $0<\lambda<1$. Hadwige…

Let $S(n)$ denote the largest edge length of a square that can be covered by $n$ unit squares, and let $\Pi(2)$ denote the minimum number of unit squares needed to cover a square o…

Let $H$ be a $k$-uniform hypergraph. For $1 \leq m \leq k-1$, let $\nu^{(m)}(H)$ be the maximum size of a set of edges of $H$ whose pairwise intersections have size less than $m$,…

Let $n$ be a positive integer, and let a convex body be a compact convex set with nonempty interior in $n$-dimensional Euclidean space. Hadwiger's covering conjecture. Every $n$-di…

NP-hardness conjecture. $C$-$k$-COVER is NP-hard for every convex body $C$ and every integer $k\geq 3$. Similarly, $C$-$k$-PACK is NP-hard for every convex body $C$ and every integ…

Let ${\cal M}=(E,\mathcal{I})$ be a matroid. It is $k$-coverable if its ground set $E$ can be covered by at most $k$ independent sets from $\mathcal{I}$. A partition matroid on the…

Zone and component inradius conjecture. Then

Cap covering inradius conjecture. The sum of the inradii of all domains in the collection is at least $\alpha$.

Three-cover multiplicity conjecture.

Restricted almost-cover conjecture. Under these assumptions,

Almost-cover conjecture. For an arbitrary fixed integer $k 1$ and sufficiently large $n$,

Let $N$ be a positive integer, let $x_1,\ldots,x_N$ be points on the unit sphere $\mathbb S^{2}$, and let $w>0$. For a unit vector $v$, consider the strip…

Let $N$ be a positive integer and let $x_1,\ldots,x_N$ be points in the unit disk in the plane. A strip of width $1/N$ is a planar strip with that width. Planar strip occupancy con…

Let $N$ be a positive integer and let $x_1,\ldots,x_N$ be points on the unit sphere $\mathbb S^{2}$. A strip of width $1/N$ in direction $v$ is the set of points satisfying…

Covering-radius conjecture. The expected covering radius satisfies

One-local-maximum-per-way conjecture. There is at most one local maximum per way.