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 Pₙ(z) = ∏ⱼ₌₁ⁿ (1 - zsⱼ), where s₁, , sₙ are positive integers. Of particular interest is the quantity f(n) = infs₁,,sₙ≥ 1 max|z|=1 |Pₙ(z)|.They proved that limn→∞ f(n)1/n = 1, and also established the classical lower bound f(n) ≥ √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) ≥ 2√n.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.