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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.