Experimental Mathematics

The Symmetric Subset Problem in Continuous Ramsey Theory

Greg Martin and Kevin O'Bryant

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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


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