## Real Analysis Exchange

### Uniformly antisymmetric functions and K5

Krzysztof Ciesielski

#### 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

https://projecteuclid.org/euclid.rae/1341343230

Mathematical Reviews number (MathSciNet)
MR1377524

Subjects
Primary: 26A16: Lipschitz (Hölder) classes 04A20