## A question about modal logic

September 5, 2008

Say that a class of Kripke frames, $\mathfrak{C}$, is compact just in case, for every set of propositional modal formulae, $\Gamma$, $\Gamma$ has a model in $\mathfrak{C}$ (i.e. a model based on a frame in $\mathfrak{C}$) iff every finite subset of $\Gamma$ has a model in $\mathfrak{C}$.

We know some sufficient conditions for $\mathfrak{C}$ to be compact, for example, if $\mathfrak{C}$ is closed under taking ultraproducts. Does anyone know any necessary conditions (or even necessary and sufficient conditions) for compactness in this sense?