Say that a set $S\subseteq\Bbb N$ is pairwise coprime if every two elements of $S$ are relatively prime. Denote by $f(n)$ the size of a maximal pairwise coprime subset of $\{1,...,n\}$.
What is the asymptotic behavior of $f(n)$ as $n\rightarrow\infty$?
I think I can prove that $f(n)=o(n)$, but that's about what I know.
No context for this question, just something I was thinking about today.
Thanks in advance.
For every prime $p\le n$, there is at most $1$ number in $S$ divisible by $p$. So taking $S$ to be $1$ together with all the primes $\le n$ maximizes the size of $S$ subject to pairwise coprimality. Now the Prime Number Theorem gives us the asymptotics.
Remark: One can characterize all maximum-sized subsets $S$ of $\{1,2,\dots,n\}$ such that any two distinct members of $S$ are coprime. Such an $S$ must consist of $1$, and for each prime $p\le n$, some power of $p$ that is $\le n$.