**Background**



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> This question follows up on the previous question I had asked [here](https://mathoverflow.net/questions/222793/how-large-can-a-subset-of-1-ldots-n-be-if-all-pairwise-lcms-of-its-elemen) That question had a nice answer, for the specific parameter choice $M=N+1,$ now I'm interested in $M=N\log N.$
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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 $M,N$ as $N\rightarrow \infty$ and $M=M(N)$ also tends to infinity.


Specifically, let $M=N \log N$. What is the maximum order of $|A|$?



Maximum possible size of subset of $\{1,\ldots,N\}$ with LCM of members bounded below
You cannot do much better then in the construction of Woett. To see this note that the size of $A$ is bounded above by the maximal size of a set $A$, such that $gcd(a, a')<\frac{N}{\log N}$ for all $a\neq a'$ in $A$. Suppose that $A$ is a set with the latter property. For each $a\in A$, let $d(a)$ be the largest divisor of $a$ which is $<\log N$. Then the map $a\mapsto\frac{a}{d(a)}$ is injective. An integer $n$ in the image of this map has the property that it is not divisible by a prime $<\frac{n}{N/\log N}$, so we obtain
$$
|A|\leq \frac{N}{(\log\log N)^2} + \#\left\{n\leq N: P^-(n)>\frac{\log N}{(\log\log N)^2}\right\}\ll\frac{N}{\log\log N}.
$$


