On sets of integers, none of which divides the product of \(k\) others (Q627932)

For an integer \(k\geq 1\) let \(F_k(x)\) be the maximal number of elements of a set \({\mathcal A}\subset {\mathbb N}\cap [1,x]\) such that there are no distinct elements \(a,a_1,\dots,a_k\in {\mathcal A}\) with \(a\) dividing \(a_1a_2\dots a_k\). The author proves that there exist absolute constants \( c_1,c_2>0\) such that, for any real number \(x>e^{48}\) and any integer \(2\leq k\leq \frac 16\sqrt{\frac{\log x}{\log\log x}}\), \[ \pi(x)+\frac{c_2}{(k+1)^2}\frac{x^{\frac{2}{k+1}}}{(\log x)^2}\leq F_k(x)\leq \pi(x)+c_1(k+1)^2\frac{x^{\frac{2}{k+1}}}{(\log x)^2}. \]