Sidon–Ramsey and $$B_{h}$$-Ramsey numbers

• Published in: Boletín de la Sociedad Matemática Mexicana 2024-11, Vol.30 (3), Article 104
• Main Authors: Espinosa-García, Manuel A., Montejano, Amanda, Roldán-Pensado, Edgardo, Suárez, J. David
• Format: Article
• Language: English
• ISSN: 1405-213X; 2296-4495
• DOI: 10.1007/s40590-024-00676-7

For a given positive integer k , the Sidon–Ramsey number $${{\,\textrm{SR}\,}}(k)$$ 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 $$B_h$$ B h set. The second generalization considers the scenario where the interval [1,  n ] is substituted with a d -dimensional box of the form $$\prod _{i=1}^d[1,n_i]$$ ∏ i = 1 d [ 1 , n i ] . For the general case of $$h\ge 3$$ 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.