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 $$f(N)\geqslant cN^{\frac{1}{3}}$$ f ( N ) ⩾ c N 1 3 , for a constant $$c>0$$ c > 0 . In this note, we improve this bound to $$N^{\frac{1}{3}+\frac{7}{78}+o(1)}$$ N 1 3 + 7 78 + o ( 1 ) .