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New sets (typically found by computer search) with Sidon constant equal tothe square root of their cardinalities are given. For each integer $N$ thereare only a finite number of groups of prime order containing $N$-elementextreme sets. Some extreme sets appear to fit a pattern; others do not. Variousconjectures and questions are given.

A beastiary of sets having extremal Sidon constant, or, there must be more than one theorem somewhere here