Editing Freiling's axiom of symmetry (section)

==Connection to graph theory==
Using the fact that in ZFC, we have <math>2^{\kappa}=\kappa^{+}\Leftrightarrow\neg\texttt{AX}_{\kappa}\,</math> (see [[Freiling's axiom of symmetry#Relation to the (Generalised) Continuum Hypothesis|above]]), it is not hard to see that the ''failure'' of the axiom of symmetry&nbsp;— and thus the success of <math>2^{\kappa}=\kappa^{+}\,</math>&nbsp;— is equivalent to the following combinatorial principle for graphs:

:* The [[complete graph]] on <math>\mathcal{P}(\kappa)\,</math> can be so directed, that every node leads to at most <math>\kappa\,</math>-many nodes.

In the case of <math>\kappa=\aleph_{0}\,</math>, this translates to:

:* The complete graph on the unit circle (or any set of the same size as the reals) can be so directed, that every node has a path to at most countably-many nodes.

Thus in the context of ZFC, the failure of a Freiling axiom is equivalent to the existence of a specific kind of choice function.