Abstract

We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a single \-sentence \ that codes Kurepa trees to prove the following statements: The spectrum of \ is consistently equal to \ and also consistently equal to \\), where \ is weakly inaccessible.The amalgamation spectrum of \ is consistently equal to \ and \\), where again \ is weakly inaccessible. This is the first example of an \-sentence whose spectrum and amalgamation spectrum are consistently both right-open and right-closed. It also provides a positive answer to a question in Souldatos :533–551, 2014).Consistently, \ has maximal models in finite, countable, and uncountable many cardinalities. This complements the examples given in Baldwin et al. :545–565, 2016) and Baldwin and Souldatos :444–452, 2019) of sentences with maximal models in countably many cardinalities.Consistently, \ and there exists an \-sentence with models in \, but no models in \. This relates to a conjecture by Shelah that if \, then any \-sentence with a model of size \ also has a model of size \. Our result proves that \ can not be replaced by \, even if \.