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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