6 problems

Let $P \subset {\mathbb R}^d$ be a finite point set of size $n$, where $d=2$ or $d=3$. Write $|p-p'|$ for the Euclidean distance between points $p,p'\in P$. Integer-distance conjec…

Let $\mathcal{R}_1$ and $\mathcal{R}_2$ be Rectangulation states, and let $H$ and $V$ denote respectively an all-horizontal and an all-vertical state. Let…

Let $\mathcal{R}_1$ and $\mathcal{R}_2$ be Rectangulation states, and let $d(\mathcal{R}_1,\mathcal{R}_2)$ denote their distance under the flip operation, while…

Let $n$ points be given in the Euclidean plane, and consider the number of occurrences of any one distance among pairs of these points. Erdős's single-distance conjecture. A single…

Let $n$ points be given in the Euclidean plane. Erdős's distinct-distances conjecture. They determine at least … distinct distances. The conjecture was essentially established by G…

Erdős's single-distance conjecture. If $\Gamma$ is the unit circle, the estimate holds with $N^{4/3}$ replaced on the right-hand side by $N\log(N)$.