2025/09/17 by Tang, Quanyu
#26D05 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 30C10
paper · doi:10.48550/arxiv.2509.14182
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.