Guy, Richard K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section C9 "Packing sums of pairs", p. 176: "Cameron & Erdős ask for an estimate of $F(n)$, the number of Sidon sequences whose members are at most $n$. With $m$ as above, it is not even known if $F(n)/2^m\to\infty$, only that the upper limit is infinite. They believe that $F(n)<n^{\epsilon\sqrt n}$. Progress has been made by Alon and by Calkin & Thomson, who showed that $|F(n)|=O(2^{n/2+o(n)})$", where $m$ is the largest size of a Sidon subset of $\{1,\ldots,n\}$. The last quoted sentence concerns sum-free sets: Guy's following paragraph credits the same papers by Alon and by Calkin with $O(2^{n/2+o(n)})$ sum-free subsets, and as a bound on Sidon sets it would be vacuous beside $A(N)\leq N^{(1/2+o(1))\sqrt N}$. The Lev--Schoen bounds that the section then records concern sum-free subsets of $\mathbb{Z}_p$, not Sidon sets. Library home: Guy 2004.

Kohayakawa, Yoshiharu and Lee, Sang June and Rödl, Vojt\v ech and Samotij, Wojciech, The number of Sidon sets and the maximum size of Sidon sets contained in a sparse random set of integers. Random Structures Algorithms (2015), 1-25.

Saxton, David and Thomason, Andrew, Hypergraph containers. Invent. Math. 201 (2015), 925-992.