On strong Sidon sets of integers

Author(s)

Kohayakawa, Yoshiharu ^[1] ; Lee, Sang June ^[2] ; Moreira, Carlos Gustavo ^{[3, 4]} ; Rodl, Vojtech ^[5]

Affiliation

^[1] Univ Sao Paulo, Inst Matemat & Estat, Rua Matao 1010, BR-05508090 Sao Paulo - Brazil

^[2] Kyung Hee Univ, Dept Math, Seoul - South Korea

^[3] IMPA, Estr Dona Castorina 110, BR-22460320 Rio De Janeiro - Brazil

^[4] Nankai Univ, Sch Math Sci, Tianjin 300071 - Peoples R China

^[5] Emory Univ, Dept Math, Atlanta, GA 30322 - USA

• Document type: Journal article
• Source: JOURNAL OF COMBINATORIAL THEORY SERIES A; v. 183, OCT 2021.
• Web of Science Citations: 0

Abstract

A set S subset of N of positive integers is a Sidon set if the pairwise sums of its elements are all distinct, or, equivalently, if vertical bar(x + w) - (y + z)vertical bar >= 1 for every x, y, z, w epsilon S with x < y <= z < w. Let 0 <= alpha < 1 be given. A set S subset of N is an alpha-strong Sidon setif vertical bar(x + w) - (y + z)vertical bar >= w(alpha) for every x, y, z, w epsilon S with x < y <= z < w. We prove that the existence of dense strong Sidon sets implies that randomly generated, infinite sets of integers contain dense Sidon sets. We derive the existence of dense strong Sidon sets from Ruzsa's well known result on dense Sidon sets {[}J. Number Theory 68 (1998), no. 1, 63-71]. We also consider an analogous definition of strong Sidon sets for sets S contained in {[}n] = [1, . . . , n], and give good bounds for F(n, alpha) = max vertical bar S vertical bar, where S ranges over all a-strong Sidon sets contained in{[}n]. (C) 2021 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 18/04876-1 - Ramsey theory, structural graph theory and applications in Bioinformatics
• Grantee: Guilherme Oliveira Mota
• Support Opportunities: Research Grants - Young Investigators Grants