An improved lower bound for Erdős--Szekeres products

In 1959, Erdős and Szekeres posed a series of problems concerning the size of polynomials of the form $$P_n(z)=\prod _{j=1}^n(1-z^{s_j}),$$ where $s_1,\dots ,s_n$ are positive integers. Of particular interest is the quantity $$f(n)=\underset{s_1,\dots ,s_n\ge 1}{\inf }\underset{\mathrm{\mid }z\mathrm{\mid }=1}{\max }\mathrm{\mid }{P}_n(z)\mathrm{\mid }\mathrm{.}$$They proved that ${\lim }_{n\to \mathrm{\infty }}f(n{)}^{1\mathrm{/}n}=1$, and also established the classical lower bound $f(n)\ge \sqrt{2n}$. However, despite extensive effort over more than six decades, no stronger general lower bound had been established. In this paper, we obtain the new bound $$f(n)\ge 2\sqrt{n}\mathrm{.}$$This gives the first improvement of the classical lower bound for the Erdős–Szekeres problem in the general case since 1959. In particular, our result confirms a remark of Billsborough et al., who observed that if the original Erdős–Szekeres proof could be fixed, the O’Hara–Rodriguez bound would yield exactly this inequality.