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 set.
- there is a definable set-satisfiable $L_{\text{Ord},\omega}$-theory, which has no definable class model.
The preprint will be ready in a few weeks I expect. I'll post a link later, when it is available.