Real Analysis Exchange

Uniformly antisymmetric functions and K5

Krzysztof Ciesielski

Full-text: Open access

Abstract

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 \(f\colon\mathbb{R}\to\mathbb{N}\).

Article information

Source
Real Anal. Exchange, Volume 21, Number 1 (1995), 147-153.

Dates
First available in Project Euclid: 3 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1341343230

Mathematical Reviews number (MathSciNet)
MR1377524

Subjects
Primary: 26A16: Lipschitz (Hölder) classes 04A20
Secondary: 03E50: Continuum hypothesis and Martin's axiom [See also 03E57]

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

Citation

Ciesielski, Krzysztof. Uniformly antisymmetric functions and K 5. Real Anal. Exchange 21 (1995), no. 1, 147--153. https://projecteuclid.org/euclid.rae/1341343230


Export citation

References

  • K. Ciesielski, On range of uniformly antisymmetric functions, Real Analysis Exchange, 19(2) (1993–94), 616–619.
  • K. Ciesielski, L. Larson, Uniformly antisymmetric functions, Real Analysis Exchange, 19(1) (1993–94), 226–235.
  • P. Kostyrko, There is no strongly locally antisymmetric set, Real Analysis Exchange, 17 (1991–92), 423–425.
  • P. Komjáth and S. Shelah, On uniformly antisymmetric functions, Real Analysis Exchange, 19(1) (1993–94), 218–225.
  • Brian Thomson, Symmetric Properties of Real Functions, Marcel Dekker, 1994.