## On the minimum diameter of plane integral point sets

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Kurz, Sascha

Wassermann, Alfred

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Since ancient times mathematicians consider geometrical objects with integral
side lengths. We consider plane integral point sets $\mathcal{P}$, which are
sets of $n$ points in the plane with pairwise integral distances where not all
the points are collinear.
The largest occurring distance is called its diameter. Naturally the question
about the minimum possible diameter $d(2,n)$ of a plane integral point set
consisting of $n$ points arises. We give some new exact values and describe
state-of-the-art algorithms to obtain them. It turns out that plane integral
point sets with minimum diameter consist very likely of subsets with many
collinear points. For this special kind of point sets we prove a lower bound
for $d(2,n)$ achieving the known upper bound $n^{c_2\log\log n}$ up to a
constant in the exponent.
A famous question of Erd\H{o}s asks for plane integral point sets with no 3
points on a line and no 4 points on a circle. Here, we talk of point sets in
general position and denote the corresponding minimum diameter by
$\dot{d}(2,n)$. Recently $\dot{d}(2,7)=22 270$ could be determined via an
exhaustive search.

Comment: 12 pages, 5 figures

Comment: 12 pages, 5 figures

##### Keywords

Mathematics - Combinatorics, 52C10 (Primary), 11D99, 53C65 (Secondary)