Freiling's Axiom of Symmetry - Connection To Graph Theory

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:

  • 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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