558 problems

The case of unit spheres in three-dimensional Euclidean space concerns packings of congruent spheres, whose density is the proportion of space they occupy. Kepler's conjecture. The…

Circle patterns are configurations of circles with prescribed combinatorial structures. A conformal mapping is a structure-preserving map between planar domains that preserves angl…

Quadratic-hole conjecture. The quadratic upper bounds for the minimum numbers of 5-holes and 6-holes are tight:

Consider a tiling of $n$-dimensional Euclidean space by identical hypercubes. Keller's conjecture. Two of the hypercubes in the tiling have an $(n-1)$-dimensional face in common. T…

Borsuk's conjecture.

Erdős–Szekeres conjecture.

Dirac's conjecture. For $|S|$ sufficiently large,

Hadwiger–Debrunner's conjecture. For every $p\geqslant q\geqslant d+1$, there exists a constant $c=c(p,q;d)$ such that every family of compact, convex sets in $\mathbb{R}^d$ with t…

Let $\mathcal{X}$ be a set of $n$ non-collinear points in the plane, and let $t(\mathcal{X})$ denote the maximum number of lines spanned by $\mathcal{X}$ that are incident with a p…

Let $n$ points form a convex polygon, and for each point count the distinct distances it determines to the other points. Erdős's convex-polygon pinned-distance conjecture. There is…

Let $g(k)$ be the maximum cardinality of a finite set $X\subset\mathbb{R}^2$ determining at most $k$ distinct distances, and call a set attaining this maximum an extremizer. Erdős–…

A planar graph is a graph embeddable in the plane, and its combinatorial curvature is the angle-deficiency curvature determined by an embedding in a surface. Higuchi's conjecture.…

Let $d>1$ and let $cOmegacsubset cmathbb{Z}^d$ be a finite set. Lagarias–Wang's conjecture. If $cOmega$ tiles $cmathbb{Z}^d$ by translation, then it admits a periodic tiling. Here…

Let $N>1$ and $3\leq d\leq 41$. Consider an arbitrary TS-packing of $N$ unit balls in $\mathbb{E}^d$. Fejes Tóth's sausage conjecture. The volume of their convex hull is at least t…

Langerman's conjecture. For any $nd$ masses on $\mathbb{R}^d$, there exist $n$ hyperplanes whose induced chessboard partition is balanced for every one of the $nd$ masses.

For a finite set of $n$ points in $\mathbb{E}^d$, its diameter graph is the graph whose vertices are the points and whose edges join pairs at the diameter of the set. A clique of s…

Fejes Tóth's conjecture. This energy is maximal when $x_1,\dots,x_{d+1}$ are mutually orthogonal and $x_k=x_{k-(d+1)}$ for every $k$ with $d+1<k\le N$. Equivalently, periodically r…

Let $T$ be a tile in $\mathbb{Z}^n$. A tiling by translated copies of $T$ is periodic if it is invariant under a nonzero translation of $\mathbb{Z}^n$. Periodic tiling conjecture.…

Let $e(d,n)$ denote the maximum number of diameters among sets of $n$ points in $\mathbb{R}^d$. For $d=3$, this is the Vázsonyi problem. Vázsonyi's conjecture. Vázsonyi conjectured…

Pach's decomposition conjecture. For every convex disk $K$, there exists a minimal natural number $m(K)$ such that every $m(K)$-fold covering of the plane by translates of $K$ can…

Let $d+1$ sets, called colours, each consist of $d+1$ points in general position in $\mathbb{R}^d$, and let $p$ be any point in the convex hull of each set. Colourful simplicial de…

Erdős distance conjecture for well-distributed sets. One has

Kusner's conjecture.

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

Erdős–Purdy–Straus conjecture. This construction is tight for all sufficiently large $n$. The conjecture asks whether the displayed upper bound gives the exact minimum once $n$ is…