Sidon set systems

TL;DR

This paper investigates the maximum size of Sidon systems of k-subsets of {1,...,N}, providing upper and lower bounds, precise results for small k, and probabilistic thresholds for random systems to be Sidon.

Contribution

It establishes new bounds on the size of Sidon systems, especially for small k, and determines the probability thresholds for random systems to be Sidon.

Findings

Upper bound: F_k(N) ≤ {N-1 choose k-1} + N - k

Asymptotic lower bound: F_k(N) = Ω_k(N^{k-1})

Threshold probability for random Sidon systems for k ≥ 2

Abstract

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 $F_k(N)\le {N-1\choose k-1}+N-k$ and the asymptotic lower bound $F_k(N)=\Omega_k(N^{k-1})$. 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 $k\ge 2$.