On strong infinite Sidon and Bh sets and random sets of integers

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:182

Article

Fabian, David¹  Rue, Juanjo^2,3,4,5  Spiegel, Christoph⁶ 

[1] Free Univ Berlin, Dept Math & Comp Sci, Berlin, Germany

[2] Univ Politecn Cataluna, Dept Math, Barcelona, Spain

[3] Univ Politecn Cataluna, Inst Math IMTech, Barcelona, Spain

[4] Ctr Recerca Matemat, Bellaterra, Spain

[5] Barcelona Grad Sch Math BOSMath, Barcelona, Spain

[6] Zuse Inst Berlin, Interact Optimizat & Learning, Berlin, Germany

关键词: Additive combinatorics;    Extremal combinatorics;    Probabilistic combinatorics;    Random sets;   

DOI  :  10.1016/j.jcta.2021.105460

来源: Elsevier

【 摘 要 】

A set of integers S subset of N is an alpha-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on alpha, more specifically if vertical bar(x + w) - (y + z)vertical bar >= max{x(alpha), y(alpha), z(alpha), w(alpha)} for every x, y, z, w is an element of S satisfying max{x, w} not equal max{y, z}. We obtain a new lower bound for the growth of alpha-strong infinite Sidon sets when 0 <= alpha < 1. We also further extend that notion in a natural way by obtaining the first non-trivial bound for alpha-strong infinite B-h sets. In both cases, we study the implications of these bounds for the density of, respectively, the largest Sidon or B-h set contained in a random infinite subset of N. Our theorems improve on previous results by Kohayakawa, Lee, Moreira and Rodl. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free