The structure of Sidon set systems

TL;DR

This paper characterizes the structure of Sidon set systems in abelian groups, establishing tight bounds on their maximum size and extending results to random families and higher sumsets.

Contribution

It provides a near-complete structural description of Sidon systems, closing previous size bounds gap and extending analysis to random and higher sumset cases.

Findings

Maximum size of Sidon systems is at least (1-o(1)) times the binomial coefficient nk-1.

Structural results imply tight bounds on Sidon system sizes.

Extensions to h-fold sumsets for fixed h  3 are achieved.

Abstract

A family $\mathcal{F}\subset 2^G$ of subsets of an abelian group $G$ is a Sidon system if the sumsets $A+B$ with $A,B\in \mathcal{F}$ are pairwise distinct. Cilleruelo, Serra and the author previously proved that the maximum size $F_k(n)$ of a Sidon system consisting of $k$-subsets of the first $n$ positive integers satisfies $C_k n^{k-1}\leq F_k(n) \leq \binom{n-1}{k-1}+n-k$ for some constant $C_k$ only depending on $k$. We close the gap by proving an essentially tight structural result that in particular implies $F_k(n)\geq (1-o(1))\binom{n}{k-1}$. We also use this to establish a result about the size of the largest Sidon system in the binomial random family $\binom{[n]}{k}_p$. Extensions to $h$-fold sumsets for any fixed $h\geq 3$ are also obtained.