The Symmetric Subset Problem in Continuous Ramsey Theory

Abstract

A symmetric subset of the reals is one that remains invariant under some reflection $x\mapsto c-x$. We consider, for any $0<\e \le 1$, the largest real number $\De$ such that every subset of $[0,1]$ with measure greater than $\e$ contains a symmetric subset with measure $\De$. In this paper we establish upper and lower bounds for $\De$ of the same order of magnitude: For example, we prove that $\De=2\e-1$ for $\frac{11}{16}\le\e\le1$ and that $0.59\e^2<\De<0.8\e^2$ for $0<\e\le\frac{11}{16}$.

This continuous problem is intimately connected with a corresponding discrete problem. A set $S$ of integers is called a $\Bg$ 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 $\De$ implies that every $\Bg$ set contained in $\{1,2,\dotsc,n\}$ has cardinality less than $1.30036\sqrt{gn}$. This improves a result of Green for $g\ge 30$. Conversely, we use a probabilistic construction of $\Bg$ sets to establish an upper bound on $\De$ for small $\e$