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2007 The Symmetric Subset Problem in Continuous Ramsey Theory
Greg Martin, Kevin O'Bryant
Experiment. Math. 16(2): 145-166 (2007).


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 $\e$


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Greg Martin. Kevin O'Bryant. "The Symmetric Subset Problem in Continuous Ramsey Theory." Experiment. Math. 16 (2) 145 - 166, 2007.


Published: 2007
First available in Project Euclid: 7 March 2008

zbMATH: 1209.05257
MathSciNet: MR2339272

Primary: 05D99
Secondary: 11B83 , 42A16

Keywords: continuous combinatorics , Ramsey theory , Sidon sets

Rights: Copyright © 2007 A K Peters, Ltd.


Vol.16 • No. 2 • 2007
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