Let $\mathcal{L} = \{\equiv\}$, where $\equiv$ is a binary relation symbol. How can I show that there is no such $\mathcal{L}$-sentence $\varphi$ that is true in every finite $\mathcal{L}$-structure $\mathcal{M}$ in which $\equiv^{\mathcal{M}}$ is an equivalence relation on $M$ that has equivalence class of at least $|M|/2$ elements and false in other finite $\mathcal{L}$-structures?
I think I have to use ultraproducts here in order to show it, but I don't know how to begin. I'd appreciate any help on this exercise.