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

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.