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 p1,p2,…,pk, where pi is the i-th prime. In 1962, Erdős conjectured that f(n,k)=|E(n,k)| for all n≥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 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.