Sidon-Ramsey and Bh-Ramsey numbers

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, i.e., one of the parts 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, i.e., 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 d-dimensional box of the form Pi(d)(i=1)[1,n(i)]. For the general case of h >= 3 and d-dimensional boxes, before applying our method to obtain the Ramsey-type result, we establish an upper bound for the corresponding density parameter.