The Keisler-Shelah isomorphism theorem and the continuum hypothesis

We show that if for any two elementary equivalent structures $\mathbf{M}, \mathbf{N}$ of size at most continuum in a countable language, $\mathbf{M}^{\omega}/ \mathcal{U} \simeq \mathbf{N}^\omega / \mathcal{U}$ for some ultrafilter $\mathcal{U}$ on $\omega,$ then $CH$ holds. We also provide some consistency results about Keisler and Shelah isomorphism theorems in the absence of $CH$.