Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4647863 | Discrete Mathematics | 2013 | 9 Pages |
Abstract
Let f(n,k) be the largest number of positive integers not exceeding n from which one cannot select k+1 pairwise coprime integers, and let E(n,k) be the set of positive integers which do not exceed n and can be divided by at least one of p1,p2,â¦,pk, where pi is the i-th prime. In 1962, ErdÅs conjectured that f(n,k)=|E(n,k)| for all nâ¥pk. In 1973, Choi proved that the conjecture is true for k=3. In 1988, Mócsy confirmed the conjecture for k=4. In 1994, Ahlswede and Khachatrian disproved the conjecture for k=212. In this paper we give a new proof of the following result: for nâ¥49, if A(n,4) is a set of positive integers not exceeding n such that one cannot select 5 pairwise coprime integers from A(n,4) and |A(n,4)|â¥|E(n,4)|, then A(n,4)=E(n,4). We also prove that the conjecture is false for k=211. Several open problems and conjectures are posed for further research.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
Yong-Gao Chen, Xiao-Feng Zhou,