  A set $A$ of nonnegative integers is called a Sidon set if there is no Sidon
4-tuple, i.e., $(a,b,c,d)$ in $A$ with $a+b=c+d$ and $\{a, b\}\cap \{c,
d\}=\emptyset$. Cameron and Erd\H os proposed the problem of determining the
number of Sidon sets in $[n]$. Results of Kohayakawa, Lee, R\" odl and Samotij,
and Saxton and Thomason has established that the number of Sidon sets is
between $2^{(1.16+o(1))\sqrt{n}}$ and $2^{(6.442+o(1))\sqrt{n}}$. An
$\alpha$-generalized Sidon set in $[n]$ is a set with at most $\alpha$ Sidon
4-tuples. One way to extend the problem of Cameron and Erd\H os is to estimate
the number of $\alpha$-generalized Sidon sets in $[n]$. We show that the number
of $(n/\log^4 n)$-generalized Sidon sets in $[n]$ with additional restrictions
is $2^{\Theta(\sqrt{n})}$. In particular, the number of $(n/\log^5
n)$-generalized Sidon sets in $[n]$ is $2^{\Theta(\sqrt{n})}$. Our approach is
based on some variants of the graph container method.
