The Symmetric Subset Problem in Continuous Ramsey Theory

A symmetric subset of the reals is one that remains invariant under some reflection x 7→ c − x. We consider, for any 0 < ε ≤ 1, the largest real number ∆(ε) such that every subset of [0, 1] with measure greater than ε contains a symmetric subset with measure ∆(ε). In this paper we establish upper and lower bounds for ∆(ε) of the same order of magnitude: for example, we prove that ∆(ε) = 2ε − 1 for 11 16 ≤ ε ≤ 1 and that 0.59ε2 < ∆(ε) < 0.8ε2 for 0 < ε ≤ 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 (s1, s2) ∈ S×S with s1+s2 = m; in the case g = 2, these are better known as Sidon sets. Our lower bound on ∆(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 ∆(ε) for small ε. AMS Msc (2000): • 05D99 Extremal Combinatorics, • 42A16 Fourier Series of Functions with special properties, • 11B83 Special Sequences.