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 $\mathsf{Add}(\omega, \omega_2)$ in a model of $\mathsf{CH}$. We also prove that this results is in a certain sense optimal as $\mathsf{Add}(\omega, \omega_2)$ forces that $\omega_2 \not\to (\omega_2, \omega : \omega_1)^2$.