Alon and Erdős [AlEr96] proved that $f(n) = n^{1/3+o(1)}$, and more precisely\[\frac{n^{1/3}}{(\log n)^{4/3}}\ll f(n) \ll \frac{n^{1/3}}{(\log n)^{1/3}}(\log\log n)^{1/3}.\]Vu [Vu07] improved the lower bound to\[f(n) \gg \frac{n^{1/3}}{\log n}.\]Conlon, Fox, and Pham [CFP21] determined the order of growth of $f(n)$ up to a multiplicative constant, proving\[f(n) \asymp \frac{n^{1/3}(n/\phi(n))}{(\log n)^{1/3}(\log\log n)^{2/3}}.\]

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