Sidon set systems

A family $\mathcal{A}$ of $k$-subsets of $\{1,2,\dots ,N\}$ is a Sidon system if the sumsets $A+B$, $A,B\in \mathcal{A}$ are pairwise distinct. We show that the largest cardinality $F_k(N)$ of a Sidon system of $k$-subsets of $[N]$ satisfies $F_k(N)\le \left(\genfrac{}{}{0px}{}{N-1}{k-1}\right)+N-k$ and the asymptotic lower bound $F_k(N)={\mathrm{\Omega }}_k(N^{k-1})$. More precise bounds on $F_k(N)$ are obtained for $k\le 3$. We also obtain the threshold probability for a random system to be Sidon for $k\ge 2$.