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1995/1996 Uniformly antisymmetric functions and K5
Krzysztof Ciesielski
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Real Anal. Exchange 21(1): 147-153 (1995/1996).


In \cite[Thm 2.5]{CL:Unif} and \cite[Thm 2]{CL:Unif2} it was proved that there is no uniformly antisymmetric function with two- and three-element range by showing that \(K_3\) and \(K_4\) can be embedded into a graph \(G(h)\) (defined below) for all appropriate \(h\). In this note we will answer Problem 1 from \cite{CL:Unif2} by showing that under the continuum hypothesis there exists \(h\) for which \(K_5\) cannot be embedded into \(G(h)\). In particular, the technique used in the proof that there is no uniformly antisymmetric function with three-element range cannot be used for the four-element range proof. Whether there exists a uniformly antisymmetric function with a finite range remains an open problem. The notion of a uniformly anti-Schwartz function is also defined, and it is proved that there exists a uniformly anti-Schwartz function \(f\colon\mathbb{R}\to\mathbb{N}\).


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Krzysztof Ciesielski. "Uniformly antisymmetric functions and K5." Real Anal. Exchange 21 (1) 147 - 153, 1995/1996.


Published: 1995/1996
First available in Project Euclid: 3 July 2012

MathSciNet: MR1377524

Primary: 04A20 , 26A16
Secondary: 03E50

Keywords: coloring of infinite graphs. , the Continuum Hypothesis , uniformly anti-Schwartz functions , uniformly antisymmetric functions

Rights: Copyright © 1995 Michigan State University Press

Vol.21 • No. 1 • 1995/1996
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