A Cameron and Erd\"os conjecture on counting primitive sets

TL;DR

This paper proves that the number of primitive subsets of \\{1,...,n\\} grows exponentially with n, confirming a conjecture by Cameron and Erd"os about the asymptotic behavior of such counting functions.

Contribution

It establishes the existence of the limit of the nth root of the counting function, confirming a conjecture about the growth rate of primitive subsets.

Findings

The limit of f(n)^{1/n} exists as n approaches infinity.

f(n) grows exponentially with n.

Confirms a conjecture by Cameron and Erd"os.

Abstract

Let $f(n)$ count the number of subsets of $\{1,...,n\}$ without an element dividing another. In this paper I show that $f(n)$ grows like the $n$-th power of some real number, in the sense that $\lim_{n\rightarrow \infty}f(n)^{1/n}$ exists. This confirms a conjecture of Cameron and Erd\"os, proposed in a paper where they studied a number of similar problems, including the well known "Cameron-Erd\"os os Conjecture" on counting sum-free subsets.