236 problems

Let $G$ be a decorated unicyclic graph with cycle length parameter $g$, let $R_G$ be its curvature operator, and let $w^\ast$ denote the Perron metric. Write … and let $\varphi$ be…

For integers $d\ge 2$ and $n\ge d+2$, let $F_1,\ldots,F_n\subseteq\mathbb{R}^d$ be sets, each of which is the union of exactly two disjoint, nonempty, closed convex sets. If, for e…

For each positive integer $n$, let $U(n)$ be the maximum, over all sets $P\subseteq\mathbb{R}^2$ with $|P|=n$, of the number of unordered pairs $\{p,q\}\subseteq P$ satisfying…

Does there exist an absolute constant $c>0$ such that, for every finite set $P\subset\mathbb{R}^3$ with $|P|=N$, the number of distinct pairwise distances satisfies…

Conjecture 1 (Lei and Bai). If $G$ is a Ricci-flat $5$-regular graph, then $G$ is either isomorphic to $RF_{72}^{5}$ or is a Cartesian product of the Petersen graph, the Triplex gr…

Let $n\ge 2$ and consider a packing of $n$ circular disks whose radii all lie in an interval $[1-\epsilon,1]$. Stability conjecture. There exists an $\epsilon>0$ such that, for eve…

Big-Line-Big-Clique Conjecture. For all integers $t$ and $\ell$ there is an integer $n$ such that every finite set $P$ of at least $n$ points in the plane either contains $\ell$ co…

Let $n$ satisfy $2\leq n\leq 8$, and let $2^k$ be the largest power of $2$ strictly less than $n$. A sphere packing is weakly recurrent and dense when it has the corresponding recu…

Let $n$ congruent regular unit hexagons form an edge-to-edge connected system. Hexagon polyomino area bound. The area of its convex hull is at most … The statement is presented in…

Maximum colourful simplicial depth conjecture. The maximum colourful simplicial depth of any point in the interior of the core is

Let $U$ be the constructed polygonal surface and let conditions 1–7 be the preceding geometric and arithmetic conditions imposed in the construction of a sail for a periodic contin…

Maximum-score conjecture. The maximum of $\sigma$ on $X$ is the constant

Let $0\leq k\leq d-1$ and let $S_0,S_1,\ldots,S_k$ be finite sets of points in $\mathbb R^d$, with … for $i=0,1,\ldots,k$. A common convex-hull transversal is a $k$-flat meeting ea…

Let $N$ be a natural number greater than $2$, let $d$ be a natural number strictly less than $N/2-1$, and let $n\geq N$ be another natural number. For the simplex $S^N(n)$ consider…

Let $\Delta_{\mathrm H}(n)$ denote the largest possible minimum area of a triangle determined by $n$ points in the unit square. Improved Heilbronn bound conjecture. There exists an…

Exponent conjecture for $h_k(n)$. For every fixed integer $k\ge 6$,

Let $A\subset\mathbb{R}^2$ be a finite point set. A visibility patch is a subset $W\subset\Gamma(\mathbb{R})$ of a real cubic on which the relevant pairs of points are considered,…

Let $A\subset\mathbb{R}^2$ be a finite planar point set. Two points of $A$ are visible if the open segment between them contains no other point of $A$; a mutually visible subset is…

Erdős–Falconer distance conjecture. If $|E|\geq Cq^{d/2}$ for a sufficiently large constant $C$ independent of $q$, then

Let $S_n$ be the set of $n^2$ points in $\mathbb{R}^2$ with integer coordinates $(x,y)$ satisfying $1\leq x,y<n$, and let $f_n$ be the maximum size of a subset of $S_n$ containing…

The discrete filling-area density conjecture.

Let a finite planar point configuration be a finite set of points in the plane, where the closest neighbors of a point are the points minimizing Euclidean distance from it. Karabeg…

Boundary-point conjecture. For any optimal solution with at least nine points, each edge of the unit square $[0,1]^2$ contains exactly two points.

Asymptotic ball Davenport conjecture. For $m\geq\sqrt d$,

Let $S \subset \mathbb{Z}^d$ be a bounded set, and let $\beta_{\mathbb{Z}}(S)$ denote its discrete Borsuk partition number. Let $\operatorname{conv}(S)$ be the convex hull of $S$,…