Consider the set $\{1,2,\ldots,N\}$. Let $LCM(a,a')$ denote the lowest common multiple of the integers $a,a'.$


We say that $A\subset \{1,2,\ldots,N\}$ is $M-$**good** if $LCM(a,a')\geq M,$ for all $a\neq a' \in A.$


How large can the size $|A|$ be for an $M-$good set, as a function of $M,N.$ Specifically let $N\rightarrow \infty$ and $M=M(N)$ also go to infinity.


Is it possible to have, say, $M=N+1$ and $|A|\geq c N$, for some constant $c\in(0,1)$? If not, what is the order of $|A|$?


Also, how slowly must $M$ grow with $N$ to have $|A| \geq c N$, with $c \in (0,1)$?



How large can a subset of $\{1,\ldots,N\}$ be if all pairwise LCMs of its elements are lower bounded?
You're asking for the size of a maximum independent set in the graph 
with vertices $\{1,2,\ldots,N\}$ and edges $(i,j)$ whenever $\text{lcm}(i,j) < M$.


I tried it in the case $M=N+1$ for $N$ up to $65$. It appears that the maximum
$|A| = \lceil N/2 \rceil$, which is attained when 
$A = \{x: \lceil (N+1)/2 \rceil \le x \le N \}$. 


