In 1959, Erdős and Szekeres posed a series of problems concerning the size of polynomials of the form Pₙ (z) = ₉=₁ⁿ (1 - z^sⱼ), where s₁, , sₙ are positive integers. Of particular interest is the quantity f (n) = ₒ䃑, , ₒ䂸 ₁ |ₙ|=₁ |Pₙ (z) |. They proved that ₍ 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) 2n. 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.
Quanyu Tang (Wed,) studied this question.