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

TL;DR

This paper presents new extremal sets with maximal Sidon constants, explores their properties across different groups, and discusses conjectures and open questions related to their structure and classification.

Contribution

It introduces newly found extremal sets with maximal Sidon constants, analyzes their distribution across groups, and proposes conjectures about their patterns and properties.

Findings

Finite number of prime order groups contain N-element extremal sets

Some extremal sets follow identifiable patterns, others do not

New extremal sets are identified through computer search

Abstract

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