Uniformly antisymmetric functions and K5

by

Krzysztof Ciesielski

Real Anal. Exchange 21 (1995-96), 147-153.

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 0<h<g(x). It is known that there exists a uniformly antisymmetric function f from reals to natural numbers, f:R-->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.

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