20 problems

Let $\mathfrak{M}(2,n)$ denote planar integral point sets with $n$ points. An integral point set is prime when it cannot be obtained by scaling a smaller integral point set while p…

Let $M\in\mathfrak{M}(m,n)$ and let $M_1,M_2\in M$ satisfy $|M_1M_2|=1$. A blowup of a facher set is the construction referred to by the source, and a unit distance is said to occu…

Let $\mathfrak{M}(2,n)$ denote planar integral point sets with $n$ points, and call $M$ optimal when it attains the minimum possible diameter among sets in $\mathfrak{M}(2,n)$. No-…

Strong regularity conjecture. For even dimension $m$, the graph of integral distances $\mathfrak{G}_{m,q}$ is a strongly regular graph. The conjecture predicts that the common-neig…

Sharpness conjecture. The upper bounds from Table $$ are sharp.

Collinear-subset conjecture. For $n\ge 9$, a plane integral point set with minimum diameter contains a subset of $n-1$ collinear points.

Circle conjecture. The points of a plane integral point set in semi-general position are situated on a circle.

Let $\mathcal{I}(n,2)$ denote the maximum number of points in an integral point set over $\mathbb{Z}_n^2$. The paper gives two lower bounds for this quantity, one in Lemma 1 and on…

An $n_m$-cluster is an integral point set in general position consisting of $n$ points in the integer grid $\mathbb{Z}^m$. Erdős–Noll conjecture. For any $m>1$ and $n>1$, there exi…

For a prime $p$, let $\dot{\mathcal{I}}(\mathbb{Z}_p,2)$ denote the maximum cardinality of an integral point set in the plane over $\mathbb{Z}_p$ in general position. Unboundedness…

Let $\mathbb{F}_q$ be a finite field, let $\square_q$ denote the set of squares in $\mathbb{F}_q$, and let $\overline{\mathcal{I}}(\mathbb{F}_q,2)$ denote the normalized maximum si…

Let $p$ be a prime and $r$ a positive integer. An integral point set in $\mathbb{Z}_{p^r}^2$ is a set of points whose pairwise squared distances are quadratic residues in…

Unbounded-minimum conjecture. For each $w\in\mathbb{N}$ there exists $q_w\in\mathbb{N}$ such that

Cardinality-spectrum conjecture. There exist $l_q,r_q\in\mathbb{N}$ such that

Let $R$ have only prime factors $p$ satisfying $p\equiv 1\pmod 4$, and let $\operatorname{circle}(R,8)$ be the integral point set obtained from the scaling construction by taking a…

Let $R$ have only prime factors $p$ satisfying $p\equiv 1\pmod 4$. Consider the two plane integral point sets supplied by the circle constructions: $\operatorname{circle}(R)$, with…

Let $gh$ and $g$ be integers for which the preceding semi-crab construction defines the plane integral point set $\mathcal{P}=\operatorname{decompose}(gh,g)$. Semi-crab maximality…

Let $h$ be an integer, and let $\mathcal{P}=\operatorname{decompose}(h)$ be the plane integral point set constructed as a crab of the order specified in the source. Decompose-crab…

Let $\mathcal{P}\subset\mathbb{Z}^m$ be a finite set of points such that the distances between any two points of $\mathcal{P}$ are integers divisible by an integer $k$. Scaling con…

Let $\mathcal{P}\subset\mathbb{Q}^m$ be a finite set of points such that the distances between any two points of $\mathcal{P}$ are integers. Embedding conjecture. There is a Euclid…