Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia, USA
A set A of integers is a Sidon set if all the sums a1+a2, with a1≤a2 and a1, a2∈A, are distinct. In the 1940s, Chowla, Erdős and Turán showed that the maximum possible size of a Sidon set contained in [n]={0,1,…,n-1} is √n (1+o(1)). We study Sidon sets contained in sparse random sets of integers, replacing the ‘dense environment’ [n] by a sparse, random subset R of [n].
Let R=[n]m be a uniformly chosen, random m-element subset of [n]. Let F([n]m)=max {|S| : S⊆[n]m Sidon}. An abridged version of our results states as follows. Fix a constant 0≤a≤1 and suppose m=m(n)=(1+o(1))na. Then there is a constant b=b(a) for which F([n]m)=nb+o(1) almost surely. The function b=b(a) is a continuous, piecewise linear function of a, not differentiable at two points: a=1/3 and a=2/3; between those two points, the function b=b(a) is constant. This is joint work with Yoshiharu Kohayakawa and Vojtech Rödl.