We know that from prime number theorem that the number of primes below $n$ is approximately $$\frac{n}{\log\_en}.$$


**$\star$** Given $n,m$, what is the largest list of pairwise coprime numbers that one can come up above $n$ and below $m$? Will it be asymptotically same as number of primes or something different? In particular can we replace $\log\_en$ by something smaller? If so, how much smaller?



Greatest number of coprime numbers between two numbers
For a given $n$, let $P\subset\{2,3\dots,n\}$ be the set of primes up to $n$, and let $S\subset\{2,3\dots,n\}$ be any subset with pairwise coprime elements. Consider the function $f:S\to P$ that assigns to any $s\in S$ its smallest prime factor $p\in P$. It is clear that $f$ is an injection, whence $|S|\leq|P|$. 


In short, the set of primes up to $n$ is the largest subset of $\{2,3,\dots,n\}$ with the given property.


**Added 1.** Péter Komjáth kindly called my attention to a [1962 survey by Paul Erdős](http://renyi.hu/~p_erdos/1962-23.pdf) (in Hungarian), which discusses some related problems. In particular, Problem 4 can be solved by the argument above: if $1\leq a\_1<\dots<a\_k\leq n$ are such that no $a\_i$ divides the product of the other $a\_j$'s, then $k\leq\pi(n)$. Problems 10 and 11 are based on a [1938 paper of Erdős](http://www.renyi.hu/~p_erdos/1938-07-Jav.pdf) and some later developments. Most relevant is Problem 18 that discusses the OP's question in general intervals. He denotes by $F(n,k)$ the maximal size of a subset $S\subset\{n+1,\dots,n+k\}$ with pairwise coprime elements, and he mentions that estimating this quantity is a difficult problem. In particular, he says that he is far from solving completely the problem of determining or estimating $\max\_n F(n,k)$. I am sure that digesting the vast [Erdős archive](http://www.renyi.hu/~p_erdos/Erdos.html) would bring up many related questions and results, in particular estimates for $F(n,k)$.


**Added 2.** The problem of estimating $F(n,k)$ (see previous section) is discussed in more detail in this [1971 paper of Erdős](http://www.renyi.hu/~p_erdos/1971-03.pdf). See also the relevant [OEIS entry](http://oeis.org/A062575).


