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: 

 -  A. Enayat and J. D. Hamkins, [ZFC proves that the class of ordinals is not weakly compact for definable classes](http://jdh.hamkins.org/ord-is-not-definably-weakly-compact/), manuscript under review. ([arχiv](https://arxiv.org/abs/1610.02729), [blog post](http://jdh.hamkins.org/ord-is-not-definably-weakly-compact/))