## Experimental Mathematics

- Experiment. Math.
- Volume 16, Issue 2 (2007), 145-166.

### The Symmetric Subset Problem in Continuous Ramsey Theory

Greg Martin and Kevin O'Bryant

#### 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 $\e$

#### Article information

**Source**

Experiment. Math., Volume 16, Issue 2 (2007), 145-166.

**Dates**

First available in Project Euclid: 7 March 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1204905872

**Mathematical Reviews number (MathSciNet)**

MR2339272

**Zentralblatt MATH identifier**

1209.05257

**Subjects**

Primary: 05D99: None of the above, but in this section

Secondary: 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series {For automorphic theory, see mainly 11F30} 11B83: Special sequences and polynomials

**Keywords**

Ramsey theory continuous combinatorics Sidon sets

#### Citation

Martin, Greg; O'Bryant, Kevin. The Symmetric Subset Problem in Continuous Ramsey Theory. Experiment. Math. 16 (2007), no. 2, 145--166. https://projecteuclid.org/euclid.em/1204905872