#### Document Type

Article

#### Publication Date

1995

#### College/Unit

Eberly College of Arts and Sciences

#### Department/Program/Center

Mathematics

#### Abstract

A function f from reals to reals (f:**R**-->**R**) is a uniformly antisymmetric function if there exists a gage function g:**R**-->(0,1) such that |f(x-h)-f(x+h)| is greater then or equal to g(x) for every x from **R** and 0R-->**N**, (see [K. Ciesielski, L. Larson, Uniformly antisymmetric functions, *Real Anal. Exchange 19 *(1993-94), 226-235]) while it is unknown whether such function can have a finite or bounded range. It is not difficult to show that there exists a uniformly antisymmetric function with an n-element range if and only if there exists a gage function g:**R**-->(0,1) such that the graph G(g) is n-vertex-colorable, where G(g) is the graph with all reals forming its vertices, and with edges being the set of all unordered pairs a,b of different reals such that |a+b|/2 < g((a+b)/2). This characterization was used to prove that there is no uniformly antisymmetric function with 3-element range by showing that G(g) contains K_{4}, the complete graph on 4 vertices, as a subgraph. (See [K. Ciesielski, On range of uniformly antisymmetric functions, *Real Anal. Exchange 19 *(1993-94), 616-619].)

In this note we show that under the continuum hypothesis there exists g for which K_{5} cannot be embedded into G(g). 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.

The notion of a uniformly anti-Schwartz function is also defined and it is proved that there exists a uniformly anti-Schwartz function f:**R**-->**N**.

#### Digital Commons Citation

Ciesielski, Krzysztof, "Uniformly Antisymmetric Functions and K5" (1995). *Faculty & Staff Scholarship*. 821.

https://researchrepository.wvu.edu/faculty_publications/821