Sidon set systems

A family A of k-subsets of {1, 2, . . . , N} is a Sidon system if the sumsets A + B, A, B is an element of 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) <= ((N-1)(k-1)) + N - k and the asymptotic lower bound F-k(N) = Omega(k)(Nk-1). More precise bounds on F-k(N) are obtained for k <= 3. We also obtain the threshold probability for a random system to be Sidon for k >= 2.