Erdős Conjecture on Sidon set density (limsup equals 1)

Establish whether there exists a Sidon subset A of the natural numbers such that limsup_{n→∞} |A ∩ {1, …, n}| / √n = 1.

Background

Sidon sets (also called B2 sets) are subsets of integers with all pairwise differences distinct. Erdős posed numerous problems on Sidon sets and conjectured a sharp asymptotic density for infinite Sidon subsets of the natural numbers.

Prior work established upper bounds: Erdős and Turán proved this limsup is at most 1 for every Sidon set, while Erdős showed examples achieving 1/2 and Krückeberg improved this to 1/√2. The conjecture asks for the full density 1 in the limsup sense.