134 problems

For a set of $n$ points in the unit disk, let the minimum area be the smallest area of a triangle determined by three of the points. Heilbronn's conjecture. This minimum is always…

Contact-number equality conjecture.

Let a finite set $X\subseteq\mathbb{R}^m$ be $2$-Ramsey if every $2$-coloring of $\mathbb{R}^m$ contains a monochromatic congruent copy of $X$. A triangle is non-equilateral if its…

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…

Let $\mathcal{F}$ be a finite family of convex sets in $\mathbb{R}^2$. The family has the $T(3)$ property if every three sets of $\mathcal{F}$ can be pierced by a line, and its lin…

Let $m\geq 1$ and $k\geq 1$ be integers. For $d=\Delta(m;k)$, let $\Delta(m;k)$ denote the minimum dimension such that any $m$ masses on $\mathbb{R}^d$ can be equipartitioned by…

Let $B_d$ be the $d$-dimensional unit ball, and let $A\subset B_d$ be a closed set containing no pair of points at distance $1$. Writing $\lambda$ for Lebesgue measure, Larman–Roge…

For a sequence of positive distances $(r_j)_{j\geq 1}$, let $\mathbf{m}_{\mathbb{R}^2}((r_j)_{j\geq 1})$ denote the supremum of the upper densities of sets in $\mathbb{R}^2$ avoidi…

Let $d$ be a dimension and let $D$ be a finite set of distances whose elements are algebraically independent over $\mathbb{Q}$. Let $\chi_{\mathbb{R}^d}(D)$ denote the chromatic nu…

Let $G$ be a finite graph. For every edge $e\in E(G)$, let $\sigma_e\in\mathcal{M}(\mathbb{R}^d)$ be an admissible measure. A copy of $G$ in $A\subset\mathbb{R}^d$ is a map…

Let $d$ be a dimension, let $D$ be a finite set of forbidden distances, and let $\chi_{\mathbb{R}^d}(D)$ be the chromatic number of the distance graph on $\mathbb{R}^d$ in which pa…

Let $d\geq 2$ and let $A\subseteq\mathbb{R}^d$ satisfy $\bar{d}(A)>0$, where $\bar{d}(A)$ denotes its upper limit density. Székely's conjecture. There is a $d_0$ such that every di…

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…

Let $I=[0,1)^d$, and let $T\subseteq \mathbb{R}^d$ be such that $I+T$ is a 2-extremal cube-tiling of $\mathbb{R}^d$, meaning a tiling satisfying the 2-extremality condition used in…

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

Let $K$ be a convex body in $\mathbb{R}^n$ and let $D$ be an integer cell. For each coordinate projection $P$ onto a coordinate subspace, let the corresponding summand of…

Zaks's finiteness conjecture. The largest neighborly family of congruent $3$-polytopes is finite.

Let $Z$ be a real vector space. For a slope $r\in\mathbb{R}$, define $\pi_r:Z\times Z\to Z$ by $\pi_r(a,b)=a+rb$, and define $\pi_\infty(a,b)=b$. For a finite collection $R$ of pro…

Let $E \subseteq \mathbb{R}^n$, with $n>1$, be a Besicovitch set, meaning that $E$ contains a unit line segment in every direction. Its upper Minkowski dimension is the upper Minko…

Let $C\subset\mathbb{R}^{2}$ be a star-shaped Jordan region, meaning a bounded planar set homeomorphic to the unit disk and containing a point $x$ such that every segment joining…

Conway–Sloane's conjecture. Tight packings in dimension $n$ decompose into parallel layers, each congruent to a tight packing in a lower dimension.

Let $m_1(\mathbb R^2)$ be the supremum of the upper densities of measurable sets $A\subseteq\mathbb R^2$ containing no two points at distance $1$. Erdős's conjecture. … The conject…

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