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SOLVED
Let $g(n)$ denote the largest $t$ such that there exist integers $2\leq a_1<a_2<\cdots <a_t <n$ such that \[P(a_1)>P(a_2)>\cdots >P(a_t)\] where $P(m)$ is the greatest prime factor of $m$. Estimate $g(n)$.
Stijn Cambie has proved (personal communication) \[g(n) \asymp \left(\frac{n}{\log n}\right)^{1/2}.\] Cambie further asks whether there exists a constant $c$ such that \[g(n) \sim c \left(\frac{n}{\log n}\right)^{1/2}.\] Cambie's proof shows that such a $c$ must satisfy $2\leq c\leq 2\sqrt{2}$.
Additional thanks to: Stijn Cambie