The number of Sidon sets and the maximum size of Sidon sets contained in a sparse random set of integers

Abstract

Abstract. 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. A well-known problem on Sidon sets is the determination of the maximum possible size F (n) of a Sidon subset of [n] = {0, 1, . . . , n − 1}. Results of Chowla, Erdős, Singer and Turán from the 1940s give that F (n) = (1 + o(1)) √ n. 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 (1))n a . We show that there is a constant b = b(a) such that, almost surely, we have F ([n]m) = n b+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 'critical' points: a = 1/3 and a = 2/3. Somewhat surprisingly, between those two points, the function b = b(a) is constant.Our approach is based on estimating the number of Sidon sets of a given cardinality contained in [n]. Our estimates also directly address a problem raised by Cameron and Erdős [On the number of sets of integers with various

On Sidon Sets in a Random Set of Vectors

“…Our proof of Theorem 2 will be provided in Subsection 3.1. The case d = 1 of Theorem 2 was also proved in [7].…”

“…For d = 1, a version of Theorem 5 was given in Lemma 3.3 in [7], but we improve the previous one as follows: (1) We have a better upper bound on Z n,1 (t) by removing the multiplicative factor ω in the base in Lemma 3.3 of [7]. (2) We remove the variable σ used in Lemma 3.3 of [7].…”

“…Our proof uses the following strategy from [7]. Let s be an integer with s < t, and let S be a seed Sidon set in [n] d of size s. For such a Sidon set S, we estimate the number of extensions of S to larger Sidon sets S * of size t containing S. Then, by summing over all Sidon sets S of size s, we will obtain an upper bound on Z n,d (t).…”

“…In order to bound the number of independent sets in G S of a given size, we will use the following result from [7].…”

“…Lemma 12 (Lemma 3.1 of [7]). For positive integers N and R and a positive real number β, let G be a graph on N vertices such that for every vertex set U with |U | ≥ R, the number e(U ) of edges in the subgraph of G induced on U satisfies…”