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 ∏ i = 1 d [ 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.