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Integral point sets over finite fields
We consider point sets in the affine plane where each
Euclidean distance of two points is an element of . These sets
are called integral point sets and were originally defined in -dimensional
Euclidean spaces . We determine their maximal cardinality
. For arbitrary commutative rings
instead of or for further restrictions as no three points on a
line or no four points on a circle we give partial results. Additionally we
study the geometric structure of the examples with maximum cardinality.Comment: 22 pages, 4 figure
On the characteristic of integral point sets in
We generalise the definition of the characteristic of an integral triangle to
integral simplices and prove that each simplex in an integral point set has the
same characteristic. This theorem is used for an efficient construction
algorithm for integral point sets. Using this algorithm we are able to provide
new exact values for the minimum diameter of integral point sets.Comment: 9 pages, 1 figur
There are integral heptagons, no three points on a line, no four on a circle
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 Erd\H{o}s.Comment: 4 pages, 1 figur
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