The number of maximum primitive sets of integers

Abstract

A set of integers is \emph{primitive} if it does not contain an element dividing another. Denote by $f(n)$ the number of maximum-size primitive subsets of $\{1,\ldots, 2n\}$. We prove that the limit $\alpha=\lim_{n\rightarrow \infty}f(n)^{1/n}$ exists. Furthermore, we present an algorithm approximating $\alpha$ with $(1+\varepsilon)$ multiplicative error in $N(\varepsilon)$ steps, showing in particular that $\alpha\approx 1.318$. Our algorithm can be adapted to estimate also the number of all primitive sets in $\{1,\ldots, n\}$. We address another related problem of Cameron and Erd\H{o}s. They showed that the number of sets containing pairwise coprime integers in $\{1,\ldots, n\}$ is between $2^{\pi(n)}\cdot e^{(\frac{1}{2}+o(1))\sqrt{n}}$ and $2^{\pi(n)}\cdot e^{(2+o(1))\sqrt{n}}$. We show that neither of these bounds is tight: there are in fact $2^{\pi(n)}\cdot e^{(1+o(1))\sqrt{n}}$ such sets.