The number of maximum primitive sets of integers

TL;DR

This paper investigates the asymptotic behavior of the number of maximum primitive subsets of integers, providing an approximation algorithm for the growth rate and refining bounds on the count of pairwise coprime sets.

Contribution

It proves the existence of the limit for the growth rate of maximum primitive sets and offers an efficient algorithm to approximate this rate, improving understanding of primitive set enumeration.

Findings

The limit  exists and is approximately 1.318.

An algorithm approximates  with multiplicative error in finite steps.

Refined bounds for the number of pairwise coprime sets in .

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.