Sidon sets and Sidon-partitions in cyclic groups through almost different sets

Abstract

We investigate the Sidon set problem in the modular case and its corresponding version in Ramsey theory. Specifically, we study the function ̅F(n) that maximizes the size of a Sidon set in Zn, as well as the minimum n such that Zn admits no Sidon r-partition (a partition whose parts are all Sidon sets), denoted by ̅SR(r), for a fixed positive integer r. We use known results and the pigeonhole principle to establish an upper bound of ̅SR(r), which allows us to find the exact values of ̅SR(r) for r ϵ {2, 3, 4, 7}. We also present a criterion for determining the non-existence of almost difference sets (ADS) in Zn. By exploiting such criterion and the connection between ADS sets and the existence of Sidon sets in Zn, we derive nontrivial upper bounds of ̅F(n) for infinitely many values of n, and we refine the upper bound on ̅SR(r) in multiple cases, determining also the exact value for r ϵ {5, 6}. Our findings shed new light on the behavior of Sidon sets in cyclic groups. In particular, we find infinitely many values for which ̅F(n) > ̅F(n + 1).