Geodesic
On diameter bounds for planar integral point sets in semi-general position
Čebyševskij sbornik, Tome 22 (2021) no. 4, pp. 344-351

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A point set $M$ in the Euclidean plane is said to be a planar integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on a straight line. A planar integral point set is said to be a set in semi-general position, if it does not contain collinear triples. The existing lower bound for mininal diameter of a planar integral point set is linear with respect to its cardinality. There were no known special diameter bounds for planar integral point sets in semi-general position of given cardinality (the known upper bound for planar integral point sets is constructive and employs planar integral point sets in semi-general position). We prove a new lower bound for minimal diameter of planar integral point sets in semi-general position that is better than linear (polynomial of power $5/4$). The proof is based on several lemmas and observations, including the ones established by Solymosi to prove the first linear lower bound for diameter of a planar integral point set.
Keywords: combinatorial geometry, diameter of a set, integral point set. 
@article{CHEB_2021_22_4_a18,
     author = {N. N. Avdeev},
     title = {On diameter bounds for planar integral point sets in semi-general position},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {344--351},
     publisher = {mathdoc},
     volume = {22},
     number = {4},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2021_22_4_a18/}
}
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N. N. Avdeev. On diameter bounds for planar integral point sets in semi-general position. Čebyševskij sbornik, Tome 22 (2021) no. 4, pp. 344-351. http://geodesic.mathdoc.fr/item/CHEB_2021_22_4_a18/