The Symmetric Subset Problem in Continuous Ramsey Theory

Abstract

A symmetric subset of the reals is one that remains invariant under some reflection z --> c-z. We consider, for any 0 < x <= 1, the largest real number D(x) such that every subset of $[0,1]$ with measure greater than x contains a symmetric subset with measure D(x). In this paper we establish upper and lower bounds for D(x) of the same order of magnitude: for example, we prove that D(x) = 2x - 1 for 11/16 <= x <= 1 and that 0.59 x^2 < D(x) < 0.8 x^2 for 0 < x <= 11/16. This continuous problem is intimately connected with a corresponding discrete problem. A set S of integers is called a B*[g] set if for any given m there are at most g ordered pairs (s_1,s_2) \in S \times S with s_1+s_2 = m; in the case g=2, these are better known as Sidon sets. Our lower bound on D(x) implies that every B*[g] set contained in \{1,2,...,n\} has cardinality less than 1.30036 \sqrt{gn}. This improves a result of Green for g >= 30. Conversely, we use a probabilistic construction of B*[g] sets to establish an upper bound on D(x) for small x.