57 problems

The strong Lonely Runner Conjecture. For every integer $d\geq1$, there exists an integer $N\geq1$ such that the equivalent bounded-time and covering properties hold; in particular,

Let $\mathcal{K}^n$ be the set of convex bodies in $n$-dimensional Euclidean space, and for $K\in\mathcal{K}^n$ let $c(K)$ be the smallest number of translations of…

Let $K\subset\mathbb{E}^d$ be a $d$-dimensional convex body. Hadwiger's covering conjecture. Every such $K$ can be covered by $2^d$ translates of its interior. This is presented as…

Makeev's conjecture. There exists an $A\in SO(n)$ such that all $n(n+1)/2$ of these $(n-1)$-planes are concurrent. The conjecture is a reformulation of Makeev's universal-cover con…

Let $[3]^d=\{0,1,2\}^d$ and let $f(d)$ denote the minimum number of proper boxes needed to double-cover $[3]^d$. Lower-bound conjecture. We conjecture that the answer to the paper'…

Let $B^d$ be the unit ball in $\mathbb{R}^d$, let $\sigma=\{v_0,\dots,v_d\}\subset B^d$ be a simplex, and let $\operatorname{conv}(\tau)^\delta$ denote the $\delta$-cylinder over…

Let $\mathcal{K}^n$ denote the set of convex bodies in $\mathbb{R}^n$, and for $K\in\mathcal{K}^n$ let $c(K)$ be the least number of translates of the interior of $K$ needed to cov…

Let $Q^d=[-1,1]^d$ be the $d$-dimensional axis-parallel cube, and let $F(K)$ denote the corresponding covering constant for a convex body $K$. Sokolin's conjecture. For every dimen…

Meylender–Thalmaier conjecture. The systolic ratio is strictly less than $1$ for every convex shape $K$, and it is maximized with respect to $K$ if and only if $K$ is a translate o…

Let $B$ be the unit disk in the plane, and for a convex shape $K\subset\mathbb{R}^2$ let $c(K\times B)$ be the length of the shortest closed generalized billiard orbit in $K$. Mall…

Let $\rho(\eps)=\big|\{x\in\Lambda:\varphi^2(x)\leq(1+\eps)\alpha_\Lambda\}\big|$. Choose $n$ such that … converges, as $\eps\downarrow0$, to a finite positive limit. Next-order as…

Hartmann et al.'s conjecture. The minimum $r$-cover problem is $NP$-hard for all algebraic values $r$ that are not unit fractions.

Let $\Gamma$ be a conical grid of order $n$. The covering number with multiplicity $k$ is the minimum number of lines required to cover every point of $\Gamma$ at least $k$ times.…

General covering conjecture. For a generic $d$-dimensional half-grid, the covering number is equal to

Levi-Hadwiger illumination conjecture. We have

Let $K_n^{(k)}$ be the complete $k$-uniform hypergraph, with its edges coloured using $r$ colours. A collection of cycles covers $(1-o(1))n$ vertices if the number of uncovered ver…

Let $C$ be a convex body in $\mathbb R^n$, and let $(1-\epsilon)C$ denote a homothetic copy of $C$ with scale factor $1-\epsilon$ for some $\epsilon>0$. Hadwiger's conjecture. Ever…

Let $f_{\max}(k)$ denote the bound obtained by optimizing the relevant function for parameter $k$, and let $f_{\max}(0)$ be its value at $k=0$. The conjecture concerns the improvem…

Let $h(n)$ be the minimum number of products of complete bipartite graphs needed to partition $E(K_4)\times E(K_n)$. Leader's covering conjecture. … The quantity $h(n)$ is an auxil…

Conway–Soifer conjecture. The triangle $T$ cannot be covered by $n^2+1$ unit equilateral triangles.

Hadwiger's conjecture. $N(K)\leqslant 2^d$ for every convex body $K\subseteq\mathbb{R}^d$. This is the classical covering problem for convex bodies; the paper presents improved bou…

Let $p_n$ be the minimal number of straight lines such that every cell of an $n\times n$ chessboard is met in its interior by at least one line. Together with Ambrus and Varga, the…

Let $K$ be an $n$-dimensional convex body. A boundary point $x$ is illuminated from a unit direction $u$ if the ray from $x$ in direction $u$ intersects the interior of $K$, and le…

Covering extension conjecture. The statement should hold not only for partitions, but also for coverings. When $B$ is the Euclidean unit ball, this recovers Kadets' theorem. The su…

Soltan's homothetic covering conjecture. Assume that $K \in \mathcal{K}^d$ and that $\lambda_1 K, \ldots, \lambda_n K$ permit a translative covering of $K$ with…