On diameter bounds for planar integral point sets in semi-general position

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.