Let $\phi$ be an $L\_{\omega\_1,\omega}$ sentence. The *amalgamation spectrum* of $\phi$ is the set of all cardinals $\kappa$ such that the models of $\phi$ of size $\kappa$ satisfy amalgamation.


**Question**: Is there a known example where the amalgamation spectrum is right-open? E.g. of the form $[\kappa,\lambda)$, where $\lambda$ is a limit cardinal.



Example with right-open amalgamation spectrum
In [this paper](https://arxiv.org/abs/1705.05821) we were able to prove that there exists some $\psi\in L\_{\omega\_1,\omega}$ and it is consistent that the amalgamation spectrum of $\psi$ is consistently equal to $[\aleph\_1,2^{\aleph\_1})$, where $2^{\aleph\_1}$ is weakly inaccessible. This (consistently) answers the question. 


