5 problems

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…

A matchstick graph is a graph drawn in the plane with each edge a straight-line segment of unit length, such that no two edges have a point in common unless that point is an endpoi…

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 $C$.

Let $G_\varepsilon$ be the graph with vertex set $\mathbb{R}^2$ in which two points are adjacent when their distance $d$ satisfies $1-\varepsilon\leq d\leq 1+\varepsilon$, and let…