Computational complexity of some Ramsey quantifiers in finite models

The problem of computational complexity of semantics for some natural language constructions – considered in [M. Mostowski, D. Wojtyniak 2004] – motivates an interest in complexity of Ramsey quantifiers in finite models. In general a sentence with a Ramsey quantifier R of the following form Rx, yH(x, y) is interpreted as ∃A(A is big relatively to the universe ∧A ⊆ H). In the paper cited the problem of the complexity of the Hintikka sentence is reduced to the problem of computational complexity of the Ramsey quantifier for which the phrase “A is big relatively to the universe” is interpreted as containing at least one representative of each equivalence class, for some given equvalence relation. In this work we consider quantifiers Rf , for which “A is big relatively to the universe” means “card(A) > f(n), where n is the size of the universe”. Following [Blass, Gurevich 1986] we call R mighty if Rx, yH(x, y) defines NP– complete class of finite models. Similarly we say that Rf is NP–hard if the corresponding class is NP–hard. We prove the following theorems: