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I'm reading this paper which says

If we have

$$ \binom n d p^{\binom d 2} = 1 $$

where $ 0 < p \le 1$, then

$$ d = 2 \log_bn - 2 \log_b \log_b n + 2 \log_b\left(\frac 1 2 e\right) + 1 + O(1) $$

where $b = \frac 1 p$.

As the authors skimmed the proof, I've completely no idea how they reached the conclusion.

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