If A⊂N is such that it does not contain a subset S consisting of k pairwise coprime integers, then we say that A has the property Pk. Let Γk denote the family of those subsets of N which have the property Pk. If Fk(n)=maxA⊂{1,2,3,…,n},A∈Γk|A| and Ψk(n) is the number of integers u∈{1,2,3,…,n} which are multiples of at least one of the first k primes, it was conjectured that Fk(n)=Ψk−1(n) for all k≥2. In this paper, we give several partial answers.