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5. Compatcness properties of graphs


    1. Add remark

      Problem 5.55.

      [Magidor] Let \phi(n) be the statement “for every graph of size \aleph_{\omega+1}, if every subgraph of size <\aleph_{\omega+1} has chromatic number \le\aleph_n, then the entire graph has chromatic number \aleph_n.”

      Is \phi(0) consistent?
          It is known that \phi(n) is consistent whenever 1\le n<\omega. In fact, the sentence \forall n\in \omega(n>0\to \phi(n)) is consistent.

          Cite this as: AimPL: From ℵ2 to infinity, available at http://aimpl.org/alephtwo.