Ali Enayat and I are currently preparing a paper whose main theorem is 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 $V$, 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 preprint is not ready yet, but I expect that it will be ready in a few weeks. I'll post a link later, when it is available.
