On a problem of Erdős on integers, none of which divides the product of \(k\) others (Q1041201)

Let \(F_{k}(N)\) denote the maximal number of integers selected from \(\{1, \dots, N\}\), so that none of them divides the product of \(k\) others. Following Erdős's approach for \(k = 2\), the authors give a lower estimation to \(F_{k}(N)\) for \(2 \leq k \leq \frac{\log N}{6\log \log N}\). They give an upper bound for \(F_{3}(N)\) and also prove that the lower bound for \(F_{3}(N)\) is sharp apart from a constant factor. The proofs use combinatorial arguments and some elementary number theory.