Complete $\mathcal{L}_{\omega_1,\omega}$-Sentences with Maximal Models in Multiple Cardinalities

In [BKS15] examples of incomplete sentences are given with maximal models in more than one cardinality. The question was raised whether one can find similar examples of complete sentences. In this paper we give examples of complete $L_{\omega_1,\omega}$-sentences with maximal models in more than one cardinality. From (homogeneous) characterizability of $\kappa$ we construct sentences with maximal models in $\kappa$ and in one of $\kappa^+, \kappa^\omega, 2^\kappa$ and more. Indeed, consistently we find sentences with maximal models in uncountably many distinct cardinalities.