$\DeclareMathOperator{\Ord}{Ord}$Let $c: \Ord^{[2]} \rightarrow 2$ be a $2$-coloring of the class of pairs of ordinals.
Is there a definable class-sized subclass $H$ of $\Ord$ which is $c$-homogeneous, that is, such that $c(H^{[2]}) = \{0\}$ or $c(H^{[2]}) = \{1\}$?