Connection To Graph Theory
Using the fact that in ZFC, we have (see above), it is not hard to see that the failure of the axiom of symmetry — and thus the success of — is equivalent to the following combinatorial principle for graphs:
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- The complete graph on can be so directed, that every node leads to at most -many nodes.
- In the case of, this translates to: The complete graph on the unit circle can be so directed, that every node leads 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.
Read more about this topic: Freiling's Axiom Of Symmetry
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