Title: Upwards Lowenheim-Skolem for $L_{\omega_1, \omega}$

Abstract: Let T be a first order theory and p a (partial) type over a countable language. Are there arbitrary large infinite models of T that omit p? In 1965 Morley showed that the answer for that is affirmative exactly when there are models of T that omit p of arbitrarily large cardinality $<\beth_{\omega_1}$.

Morley's theorem does not generalize to uncountable languages in the expected way. I will survey a couple of results (which are probably due to Shelah) about the behaviour of this problem for uncountable languages. Then, I will return to the countable case and talk about the "maximal models spectrum" (which appears implicitly in Morley's lower bound). I will show a recent result of Sinapova and Ioannis, in which a set theoretical spectrum was used in order to gain some control over this model theoretical spectrum.