Complete Bipartite Partition Relations in Cohen Extensions

We investigate the effect of adding ${\omega }_2$ Cohen reals on graphs on ${\omega }_2$, in particular we show that ${\omega }_2\to ({\omega }_2,\omega :\omega {)}^2$ holds after forcing with $\mathsf{A}\mathsf{d}\mathsf{d}(\omega ,{\omega }_2)$ in a model of $\mathsf{C}\mathsf{H}$. We also prove that this results is in a certain sense optimal as $\mathsf{A}\mathsf{d}\mathsf{d}(\omega ,{\omega }_2)$ forces that ${\omega }_2\to ̸({\omega }_2,\omega :{\omega }_1{)}^2$.