On sequences containing at most 4 pairwise coprime integers

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 p 1 , p 2 , … , p k , where p i is the i -th prime. In 1962, Erdős conjectured that f ( n , k ) = | E ( n , k ) | for all n ≥ p k . 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 n ≥ 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 ) | ≥ | 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.