Ruzsa's problem on Bi-Sidon sets

Abstract

A subset $S$ of real numbers is called bi-Sidon if it is a Sidon set with respect to both addition and multiplication, i.e., if all pairwise sums and all pairwise products of elements of $S$ are distinct. Imre Ruzsa asked the following question: What is the maximum number $f(N)$ such that every set $S$ of $N$ real numbers contains a bi-Sidon subset of size at least $f(N)$? He proved that $f(N)\ge c{N}^{\frac{1}{3}}$, for a constant $c>0$. In this note, we improve this bound to $N^{\frac{1}{3}+\frac{7}{78}+o(1)}$.

9 pages, updated literature and exposition