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: $\chi:\upharpoonright H\times H$ is constantly 0 or 1? `

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 $X$.