## 52C10 Erd}os problems and related topics of discrete geometry [See also 11Hxx]

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#### Keywords

- Kombinatorik (4)
- exhaustive search (4)
- ganzzahlige Abstände (4)
- integral distances (4)
- Durchmesser (3)
- erschöpfende Suche (3)
- orderly generation (3)
- diameter (2)
- ordnungstreues Erzeugen (2)
- Abstand (1)

#### Institute

- Mathematik (4)
- Informatik (1)

- Maximal integral point sets over Z^2 (2008)
- Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set P={p(1),...,p(n)} in Z^2 a maximal integral point set over Z^2 if all pairwise distances are integral and every additional point p(n+1) destroys this property. Here we consider such sets for a given cardinality and with minimum possible diameter. We determine some exact values via exhaustive search and give several constructions for arbitrary cardinalities. Since we cannot guarantee the maximality in these cases we describe an algorithm to prove or disprove the maximality of a given integral point set. We additionally consider restrictions as no three points on a line and no four points on a circle.

- On the minimum diameter of plane integral point sets (2007)
- Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets 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_2loglog n} up to a constant in the exponent.

- There are integral heptagons, no three points on a line, no four on a circle (2007)
- We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erdös.

- Integral point sets over Z_n^m (2007)
- There are many papers studying properties of point sets in the Euclidean space or on integer grids, with pairwise integral or rational distances. In this article we consider the distances or coordinates of the point sets which instead of being integers are elements of Z_n, and study the properties of the resulting combinatorial structures.