An improved lower bound for Erd\H{o}s--Szekeres products

Abstract

In 1959, Erd\H{o}s 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 $$ f(n) = \inf_{s_1,\dots,s_n\ge 1} \max_{|z|=1} |P_n(z)|. $$They proved that $\lim_{n\to\infty} f(n)^{1/n} = 1$, 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 $$ f(n) \ge 2\sqrt{n}. $$This gives the first improvement of the classical lower bound for the Erd\H{o}s--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 Erd\H{o}s--Szekeres proof could be fixed, the O'Hara--Rodriguez bound would yield exactly this inequality.