  For a given positive integer $k$, the Sidon-Ramsey number $\SR(k)$ is defined
as the minimum value of $n$ such that, in every partition of the set $[1, n]$
into $k$ parts, there exists a part that contains two distinct pairs of numbers
with the same sum. In other words, there is a part that is not a Sidon set. In
this paper, we investigate the asymptotic behavior of this parameter and two
generalizations of it. The first generalization involves replacing pairs of
numbers with $h$-tuples, such that in every partition of $[1, n]$ into $k$
parts, there exists a part that contains two distinct $h$-tuples with the same
sum. Alternatively, there is a part that is not a $B_h$ set. The second
generalization considers the scenario where the interval $[1, n]$ is
substituted with a non-necessarily symmetric $d$-dimensional box of the form
$\prod_{i=1}^d[1,n_i]$. For the general case of $h\geq 3$ and non-symmetric
boxes, before applying our method to obtain the Ramsey-type result, we needed
to establish an upper bound for the corresponding density parameter.
