The maximum size of a Sidon set contained in a sparse random set of integers

• Authors: Yoshiharu Kohayakawa;Sangjune Lee;Vojtěch Rödl
• Venue: Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
• Year: 2011

A set A of non-negative integers is called a Sidon set if all the sums a1 + a2, with a1 ≤ a2 and a1, a2 ∈ A, are distinct. One of the best studied problems on Sidon sets is the determination of the maximum possible size F(n) of a Sidon subset of [n] = {0, 1,..., n − 1}. Thanks to results of Chowla, Erdős and Turán from the 1940s, it is known that F(n) = (1 + o(1))n. In this paper we study Sidon subsets of sparse random sets of integers, replacing the 'dense environment' [n] by a sparse, random subset R of [n], and ask how large a subset S ⊂ R can be, if we require that S should be a Sidon set. Let R = [n]m be a random subset of [n] of cardinality m = m(n), with all the (nm) subsets of [n] equiprobable. We investigate the random variable F([n]m) = max |S|, where the maximum is taken over all Sidon subsets S ⊂ [n]m, and obtain quite precise information on F([n]m) for the whole range of m. An abridged version of our results states as follows. Let 0 ≤ a ≤ 1 be a fixed constant and suppose m = m(n) = (1 + o(1))na. We show that there is a constant b = b(a) such that, almost surely, we have F([n]m) = nb+o(1). As it turns out, the function b = b(a) is a continuous, piecewise linear function of a that is non-differentiable at two points: a = 1/3 and a = 2/3. Somewhat surprisingly, between those two points, the function b = b(a) is constant.