All Random Solved Random Open
Let $n\geq 1$ and $f(n)$ be maximal such that, for every $a_1\leq \cdots \leq a_n\in \mathbb{N}$ we have \[\max_{\lvert z\rvert=1}\left\lvert \prod_{i}(1-z^{a_i})\right\rvert\geq f(n).\] Estimate $f(n)$ - in particular, is it true that there exists some constant $c>0$ such that \[f(n) \geq \exp(n^{c})?\]
Erdős and Szekeres [ErSz59] proved that $\lim f(n)^{1/n}=1$ and $f(n)>\sqrt{2n}$. Erdős proved an upper bound of $f(n) < \exp(n^{1-c})$ for some constant $c>0$ with probabilistic methods. Atkinson [At61] showed that $f(n) <\exp(cn^{1/2}\log n)$ for some constant $c>0$.

This was improved to \[f(n) \leq \exp( cn^{1/3}(\log n)^{4/3})\] by Odlyzko [Od82].

If we denote by $f^*(n)$ the analogous quantity with the assumption that $a_1<\cdots<a_n$ then Bourgain and Chang [BoCh18] prove that \[f^*(n)< \exp(c(n\log n)^{1/2}\log\log n).\]

Additional thanks to: Zachary Chase, Stefan Steinerberger