Products of Two Integers Avoiding Perfect Powers

For integers $d\ge 3$, let $F_{2,d}(n)$ be the largest size of a subset of $[n]$ containing no two distinct elements whose product is a perfect $d$-th power, and let $f_{2,d}(n)$ denote the analogous quantity when the two elements need not be distinct. Fleiner, Juh'asz, K"ov'er, Pach, and S'andor proved that both complements have order $n^{2\mathrm{/}3}$ when $d=3$, and asked for a leading constant. They also asked whether, more generally, $n-F_{k,d}(n)$ and $n-f_{k,d}(n)$ have order $n^{k\mathrm{/}d}$ for $1<k<d$. We establish asymptotic formula in the case $k=2$ for every fixed $d\ge 3$, $$n-F_{2,d}(n)\sim n-f_{2,d}(n)\sim C_d{n}^{2\mathrm{/}d}(\log n{)}^{d-3},$$ where $C_d>0$ is given explicitly by an Euler product and a polytope volume. In particular, the extra logarithmic factor gives a negative answer to the second question for every $d\ge 4$. For $d=3$ we obtain $$C_3=\frac{{\pi }^2}{4}\prod _p(1-\frac{3}{p^2}+\frac{2}{p^3}),$$ which answers the first question. The proof uses an exact decomposition into complementary $d$-free kernel classes, a squarefree sieve in multiplicative boxes, and a two-height polytope calculation.