Number of pairwise relatively prime integers in an interval?

Question

Given a positive integer $x$, what is the size of the largest set $A$ of integers in $[1,x]$ such that $gcd(m,n)=1$ for all $m,n \in A$?

Clearly, $|A| \geq c \frac{x}{\log x}$ by considering the set of primes up to $x$. Can we do better?

What if the interval $[1,x]$ is replaced by $[x,y]$?

Answer 1

Let $A=\{a_1,a_2,\ldots ,a_k\}$ be a set with $1\le a_1 All the $a_i$'s have distinct prime divisors (since they are coprime), which means that $k\le \pi(x)+1$.(If we let the number $1$ occur among the $a_i$'s otherwise we have $k\le\pi(x)$)
So, $|A|\le \pi(x)+1$.
If we let the interval be $[x,y]$ then the problem is quite difficult ,see here