On a conjecture of Erdős about sets without k pairwise coprime integers

2017/05/16 by Sándor Z. Kiss, Kiss, Sándor Z., Csaba Sándor +3
Computer Science · Mathematics · Social Sciences · #11B75 #FOS: Mathematics #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory #Number Theory (math.NT) #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.1705.05730

openalex publication_date 2017/05/16 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28

Abstract

Let ℤ⁺ be the set of positive integers. Let C_k denote all subsets of ℤ⁺ such that neither of them contains k + 1 pairwise coprime integers and C_k(n)=C_k∩ \1,2,…,n\. Let f(n, k) = max_A ∈ C_k(n)|A|, where |A| denotes the number of elements of the set A. Let E_k(n) be the set of positive integers not exceeding n which are divisible by at least one of the primes p₁, …, p_k, where p_i denote the ith prime number. In 1962, Erdős conjectured that f(n, k) = |E(n,k)| for every n ≥ p_k. Recently Chen and Zhou proved some results about this conjecture. In this paper we solve an open problem of Chen and Zhou and prove several related results about the conjecture.

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