33 problems

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…

Let an integral circular point set be a finite set of points on a circle in the Euclidean plane such that the distance between every pair of points is an integer. Anning's conjectu…

Large minimal-distance conjecture. There is a fixed constant $c>0$ such that the minimal distance is at least

Large minimal-distance conjecture. There is a fixed constant $c>0$ such that the minimal distance is at least

Let $\mathcal{P}_{48}$ be the displayed planar integral point set of cardinality $48$, meaning a finite set of $48$ points in the plane whose pairwise distances are rational. An in…

Rails PIPS conjecture. There exists a rails PIPS of arbitrary cardinality with $2$ points on one line and all the remaining points on the other line.

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-…

Let $d(2,n)$ be the minimum diameter of a planar integral point set with $n$ points. Consider the two open square-container problems: whether there is a $\mathit{\delta}<1$ such th…

Let $d(m,n)$ denote the minimum diameter of an integral point set in $\mathfrak{M}(m,n)$, and let the approach of the stated main estimate refer to the method used to obtain the pa…

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 $\dot{\mathcal{I}}(n,2)$ denote the maximum number of points in $\mathbb{Z}_n^2$ with pairwise integral distances, with no three points collinear and no four points situated on…

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…

Let $d(m,n)$ be the minimum possible diameter of an $m$-dimensional integral point set consisting of $n$ points, and consider the bounds supplied by the paper's blow-up constructio…

Let $d(m,n)$ denote the minimum possible diameter of an $m$-dimensional integral point set consisting of $n$ points, where an integral point set has pairwise integral distances and…

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…

Minimum-size conjecture. For $q\ge11$,

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