16 problems

Harborth's conjecture. The maximum number of edges of a matchstick graph on $n$ vertices is

Swanepoel's conjecture. The maximum number of edges in a triangle-free penny graph on $n$ vertices is

Let $S^2_r$ be the 2-dimensional sphere of radius $r>1/2$, and let a three-coloring assign one of three colors to every point of $S^2_r$. Simmons's conjecture. Every such coloring…

Let $n>1$ and $m\geq 1$, and let $X\subseteq (\mathbf{R}^{n})^{m}\cong\mathbf{R}^{mn}$ be definable using $L$, with the relation $X$ preserved under isometries of $\mathbf{R}^{n}$.…

Let $\mathbb F$ be a field of characteristic $0$. For each $n\geq 2$, let $D_n(\mathbb F)$ denote the set of positive distances that are forced by finite unit-distance configuratio…

Let $n\geq 2$, let $\mathbb F$ be a field of characteristic $0$, and let a map $f:\mathbb R^n\to\mathbb F^n$ preserve unit distance, meaning that $\varphi_n(f(x),f(y))=1$ whenever…

Exponential convergence conjecture. There exist absolute constants $c>0$ and $d_0\in\mathbb{N}$ such that, for all $d\geqslant d_0$,

Let $f(n)$ denote the minimum independence number among $n$-vertex unit-distance graphs in the plane, and let $m_1(\mathbb R^2)$ be the supremum of the upper densities of measurabl…

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…

Let $u(n)$ denote the maximum number of pairs of points at unit distance among $n$ points in the Euclidean plane $\mathbb{R}^2$. Erdős's conjecture. … This is a central open proble…

Let $\chi_{f,0}(\mathbb{R}^2)$ denote the supremum of $\chi_f(G)$ over all finite unit distance graphs $G\subseteq\mathbb{R}^2$. Finitary fractional chromatic number conjecture. ……

Let $\delta_{\mathrm{Croft}}$ be Croft's density bound for measurable 1-avoiding subsets of the plane. Let $m_1(\mathbb{R}^2)$ be the measurable independence ratio of the plane, an…

Let $G$ be a finite unit distance graph in the Euclidean plane, and let $\alpha_1(G)$ denote its independence ratio. An independence ratio is attained at one quarter when…

Let $X\subseteq\mathbb{R}^d$ be a convex set with circumradius $r(X)=1$, and let its minimum enclosing ball (m.e.b.) be the smallest closed ball containing $X$. Consider the unit d…

Smiling-bouquet conjecture. For every bouquet $C$, every colouring of the plane with finitely many but at least two colours contains a smiling congruent copy of $C$.

Hadwiger–Nelson conjecture.