Show that every element in class $\mathcal{K}$ have at most $n$ elements

Question

Suppose that $\mathcal{K}$ is a class of finite structure of language $\mathcal{L}$. If $\mathcal{K}$ is axiomatizable then prove that exist $n$ such that every structure from $\mathcal{K}$ have at most $n$ elements.

Offcourse, if that class is finite then it is trivial. Otherwise, I don't know how to solve problem. I know that exist some $\mathcal{L}$-theory $T$ such that $\mathcal{K}=\{\mathbf{M} \mid \mathbf{M} \models T \}$. Also, I think, because we want something finite, we should use somehow Compactness theorem.

Answer 1

Hint: Consider the theory of $\mathcal{K}$ adjoined with the statement "the model has at least $n$ elements" for every $n \in \mathbb{N}$. Apply compactness.

Answer 2

HINT: If $T$ is a theory that has arbitrarily large finite models, then it has an infinite model.