Real Analysis Exchange

A Certain 2-Coloring of the Reals

Péter Komjáth

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There is a function $F:[\mathfrak{c}]^{\lt\omega}\to\{0,1\}$ such that if $A\subseteq [\mathfrak{c}]^{\lt\omega}$ is uncountable, then $\{F(a\cup b):a,b\in A, a\neq b\}=\{0,1\}$. A corollary is that there is a function $f:\mathbb{R}\to\{0,1\}$ such that if $A\subseteq\mathbb{R}$ is uncountable, $2\leq k\lt\omega$, then both 0 and 1 occur as the value of $f$ at the sum of $k$ distinct elements of $A$. This was originally proved by Hindman, Leader, and Strauss under CH, and they asked if it holds in general.

Article information

Real Anal. Exchange, Volume 41, Number 1 (2016), 227-232.

First available in Project Euclid: 29 March 2017

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05D10: Ramsey theory [See also 05C55] 03E05: Other combinatorial set theory

Ramsey theory coloring the reals Sierpinski’s theorem


Komjáth, Péter. A Certain 2-Coloring of the Reals. Real Anal. Exchange 41 (2016), no. 1, 227--232.

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