The counting version of a problem of Erd\H{o}s

Abstract

A set $A$ of natural numbers possesses property $\mathcal{P}_h$, if there are no distinct elements $a_0,a_1,\dots ,a_h\in A$ with $a_0$ dividing the product $a_1a_2\dots a_h$. Erd\H{o}s determined the maximum size of a subset of $\{1,\ldots, n\}$ possessing property $\mathcal{P}_2$. More recently, Chan, Gy\H{o}ri and S\'ark\"ozy solved the case $h=3$, finally the general case also got resolved by Chan, the maximum size is $\pi(n)+\Theta_h(\frac{n^{2/(h+1)}}{(\log n)^{2}})$. In this note we consider the counting version of this problem and show that the number of subsets of $\{1,\ldots, n\}$ possessing property $\mathcal{P}_h$ is $T(n)\cdot e^{\Theta(n^{2/3}/\log n)}$ for a certain function $T(n)\approx (3.517\dots)^{\pi(n)}$. For $h>2$ we prove that the number of subsets possessing property $\mathcal{P}_h$ is $T(n)\cdot e^{\sqrt{n}(1+o(1))}$. This is a rare example in which the order of magnitude of the lower order term in the exponent is also determined.