Set systems with distinct sumsets

Abstract

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