  Ramsey quantifiers are a natural object of study not only for logic and
computer science, but also for the formal semantics of natural language.
Restricting attention to finite models leads to the natural question whether
all Ramsey quantifiers are either polynomial-time computable or NP-hard, and
whether we can give a natural characterization of the polynomial-time
computable quantifiers. In this paper, we first show that there exist
intermediate Ramsey quantifiers and then we prove a dichotomy result for a
large and natural class of Ramsey quantifiers, based on a reasonable and
widely-believed complexity assumption. We show that the polynomial-time
computable quantifiers in this class are exactly the constant-log-bounded
Ramsey quantifiers.
