SOLVED
Let $N\geq 1$. What is the size of the largest $A\subset \{1,\ldots,N\}$ such that $\mathrm{lcm}(a,b)\leq N$ for all $a,b\in A$?
Is it attained by choosing all integers in $[1,(N/2)^{1/2}]$ together with all even integers in $[(N/2)^{1/2},(2N)^{1/2}]$?
Let $g(N)$ denote the size of the largest such $A$. The construction mentioned proves that
\[g(N) \geq \left(\tfrac{9}{8}n\right)^{1/2}+O(1).\]
Erdős
[Er51b] proved $g(N) \leq (4n)^{1/2}+O(1)$. Chen
[Ch98] established the asymptotic
\[g(N) \sim \left(\tfrac{9}{8}n\right)^{1/2}.\]
Chen and Dai
[DaCh06] proved that
\[g(N)\leq \left(\tfrac{9}{8}n\right)^{1/2}+O\left(\left(\frac{N}{\log N}\right)^{1/2}\log\log N\right).\]
In
[ChDa07] the same authors prove that, infinitely often, Erdős' construction is not optimal: if $B$ is that construction and $A$ is such that $\lvert A\rvert=g(N)$ then, for infinitely many $N$,
\[\lvert A\rvert\geq \lvert B\rvert+t,\]
where $t\geq 0$ is defined such that the $t$-fold iterated logarithm of $N$ is in $[0,1)$.
