The Symmetric Subset Problem in Continuous Ramsey Theory

Abstract

A symmetric subset of the reals is one that remains invariant under some reflection x → c - x. We consider, for any 0 < ε ≤ 1, the largest real number Δ(ε) such that every subset of [0, 1] with measure greater than ε contains a symmetric subset with measure Δ(ε). In this paper we establish upper and lower bounds for Δ(ε) of the same order of magnitude: For example, we prove that Δ(ε) = 2ε - 1 for and that 0.59ε² < Δ(ε) < 0.8ε² for .

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) ∈ S × S with s ₁ + s ₂ = m; in the case g = 2, these are better known as Sidon sets. Our lower bound on Δ(ε) implies that every B*[g] set contained in {1, 2, . . . , n} has cardinality less than 1.30036√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 Δ(ε) for small ε.

Keywords:

• Primary 05D99
• Secondary 42A16, 11B83
• Ramsey theory
• continuous combinatorics
• Sidon sets