TITLE: Ramsey Theorem for the class ORD
QUESTION [12 upvotes]: 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 $X$.

REPLY [12 votes]: 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, manuscript under review. (arχiv, blog post)