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Is it true that given a (definable) 2-coloring of the ORD (class of ordinals), $\chi:[ORD]^{2}\rightarrow\lbrace 0,1\rbrace$, there exists an unbounded $H\subseteq ORD$ which is homogenous, i.e., $\chi:\upharpoonright [H]^2$ is constantly 0 or 1? `

In the above $[X]^2$ is the collection of unordered pairs from $X$.

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    $\begingroup$ You presumably meant not $ORD^2$ and $H\times H$ but the classes of unordered pairs from $ORD$ and $H$. I suspect you also intended to ask about provability rather than truth. I can give you an easy answer about truth: Yes, it's true (i.e., $ORD$ really is weakly compact). Unfortunately, that's just my opinion, and provability is another matter. It's surely not provable in the usual theories of sets and classes like NBG or MK. $\endgroup$
    – Andreas Blass
    Commented Sep 30, 2016 at 20:35

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Ali Enayat and I have proved that with respect to definable classes, Ord is NOT weakly compact. In particular, we show, in every model of ZFC,

  • there is a definable Ord-tree with no definable cofinal branch.
  • there is a definable 2-coloring of a definable proper class, with no definable homogeneous proper class.
  • there is a definable set-satisfiable $L_{\text{Ord},\omega}$-theory, which has no definable class model.

This result surprised me very much, since it shows that with respect to definable classes, we can prove that Ord fails to have a large cardinal property that reasonable people might have expected to hold true.

The article is now available:

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    $\begingroup$ We can reduce from the $V$-coloring to an Ord-coloring, if we know that there is a definable bijection of $V$ with Ord, but otherwise we seem really to need $V$ rather than Ord in that part. $\endgroup$
    – Joel David Hamkins
    Commented Sep 30, 2016 at 23:32
  • $\begingroup$ So in my comment on the question, where I said "Yes, it's true," I should emphasize that I was taking the question very literally, with "definable" applying to the coloring but not to the homogeneous class $H$. $\endgroup$
    – Andreas Blass
    Commented Sep 30, 2016 at 23:38
  • $\begingroup$ I can agree with that. If you imagine that Ord comes from a large cardinal in some kind of larger universe, then that is precisely what you'd expect. $\endgroup$
    – Joel David Hamkins
    Commented Oct 1, 2016 at 0:06
  • $\begingroup$ I updated with links, now that the article is available. $\endgroup$
    – Joel David Hamkins
    Commented Oct 11, 2016 at 0:27
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    $\begingroup$ Thanks for posting the article. Could one also prove this claim in the following way: Suppose for a contradiction that ORD is weakly compact for definable sets in $M\models ZFC$. Then we can build a minimal type $p$ over $M$. Furthermore, $p$ can be used to build a conservative elementary end extension of $M$. However, this is a contradiction since models of ZFC do not have conservative elementary end extensions. $\endgroup$
    – Sam Dworetzky
    Commented Nov 4, 2016 at 19:01

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