58 problems

Let $\mathcal{H}$ be a Coxeter arrangement on a finite-dimensional inner product space $V$, let $a\in V$, and let $R$ satisfy $\|a\|\leq R$. Assume that the number of hyperplanes o…

Let $1\le k\le d$. Let $X$ be a finite transitive subset of the unit sphere in $\mathbb{C}^d$. Unrestricted cylindrical width conjecture. There is a complex $k$-dimensional subspac…

Let a coloring $\chi$ be a partition of $\mathbb{R}^2$ into two color classes. A coloring contains a triangle $T$ if there is a monochromatic copy of $T$, where copies are obtained…

Let $T$ be a triangle in the Euclidean plane, and let a coloring be a partition of $\mathbb{R}^2$ into two color classes. A coloring contains $T$ if it has a monochromatic copy of…

Let $\mathcal{L}$ be the fixed line segment, $\mathcal{S}$ the square, and let $\mathcal{T}$ denote the triangular objects in the configuration. Consider configurations with…

$(n,k)$-set dimension conjecture. If $E\subset\mathbb{R}^n$ is an $(n,k)$ set, then

Finite-field Erdős distance conjecture. If

One-point extension conjecture. The set $X\cup\{z\}$ is Ramsey.

A box in $\mathbb{R}^d$ is an axis-parallel $d$-dimensional cuboid. Two boxes are $k$-neighborly if their intersection has dimension strictly smaller than $d$ but at least $d-k$. L…

A box in $\mathbb{R}^d$ is an axis-parallel $d$-dimensional cuboid. Two boxes are $k$-neighborly if their intersection has dimension strictly smaller than $d$ but at least $d-k$. L…

For each $3$-connected embedded graph $(G,\mathbf p)$ on the unit sphere with topological degree one and with every facial perimeter of length less than $2\pi$, define a partial or…

Let $G$ be a vertex $3$-connected planar graph embedded in the unit-radius sphere. Assume that each edge is a geodesic of length less than $pi$ and each face has total perimeter le…

Let $P$ be a set in the relevant Euclidean space, and let $A$ be a Baire set. An affine image of $P$ is a set obtained from $P$ by an affine transformation. Topological Erdős simil…

Maxwell's conjecture. The electrostatic potential defined by $n$ point charges in $\mathbb{R}^3$ has at most $(n-1)^2$ equilibria.

Let $n$ be the number of pentagons in the model, and let $V(n)$, $E(n)$, and $H(n)$ denote the numbers of vertices, edges, and holes, respectively. The expectation is taken over th…

Let $P_0$ be the original pentagon in the Eden model, and let $u_1,u_2,u_3,u_4,u_5$ be the five possible translation vectors between centers of neighboring pentagons. The set of po…

Let $P$ be a set of $n$ points in the plane. Write $w(P)$ for the length of a Euclidean minimum spanning tree of $P$, and let … where the maximum is over all non-trivial bipartitio…

A geometric polyhedron here may be nonconvex and self-intersecting; its faces, dihedral angles, edge lengths, and partially-flat vertices are understood in the sense of the paper.…

Maximum segmentation conjecture. For each $k\in\mathbb N$, $k\ne 2$, there exists a ruleset of size $k$ that has a $(k+1)$-segmentation. For each $k\in\mathbb N$, there is no rules…

Let $n\ge 2$, and let $\mathcal{F}_1,\mathcal{F}_2,\dots,\mathcal{F}_n$ be families of translates of a convex compact set $\mathcal{K}$ in the plane. Assume that … for every…

Connelly's uniformity conjecture. The ternary triangulated packing numbered $53$ is the most uniform triangulated packing other than HCP. Its proposed deformation yields the highes…

Density-maximizing conjecture. For the ratios $r_b$ and $r_c$, the periodic nontriangulated binary packing depicted in Fig. 7 maximizes density among all binary packings with the s…

Consider $n$ equally spaced points on a circle and write $m=\lfloor n/2\rfloor$. A multiset is represented as $[\ell_1,\ldots,\ell_m]$, where $\ell_j$ counts chords of type $j$. Le…

For $P\subseteq\mathbb{R}^2$, let $M_P$ denote the associated finite forbidden $0$–$1$ matrix when $P$ is finite, and let $\operatorname{osat}(n,P)$ be the continuous saturation fu…

For $P\subseteq\mathbb{R}^2$, call an open set $S\subseteq[0,n]^2$ saturating for $P$ if $S$ avoids $P$ and every proper open superset of $S$ in $[0,n]^2$ contains $P$. Define…