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 $[0,1)$.