2021/11/15 by Manuel A. Espinosa-García, Espinosa-García, Manuel A., Amanda Montejano +5
Computer Science · Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2111.08076
openalex publication_date 2021/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Bh 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 ∏i=1d[1,ni]. For the general case of h≥ 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.