The structure of Sidon set systems

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 $C_k{n}^{k-1}\le F_k(n)\le \left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)+n-k$ for some constant $C_k$ only depending on $k$. We close the gap by proving an essentially tight structural result that in particular implies $F_k(n)\ge (1-o(1))\left(\genfrac{}{}{0px}{}{n}{k-1}\right)$. We also use this to establish a result about the size of the largest Sidon system in the binomial random family ${\left(\genfrac{}{}{0px}{}{[n]}{k}\right)}_p$. Extensions to $h$-fold sumsets for any fixed $h\ge 3$ are also obtained.