An existential fragment of second order logic (Q1306791)

The author deals with the set SO(\(\exists\)) of those second-order sentences in a finite relational signature, whose quantifier prefix is an arbitrary string of second-order quantifiers followed by a string of existential first-order quantifiers. This fragment of second-order logic is shown to be decidable with respect to finite satisfiability; some other model-theoretic properties of it (compactness, finite submodel property) are also discussed. Using Ramsey theory in a somewhat unusual context, the author obtains a few non-definability results for SO(\(\exists\)). At last a hierarchy of finite-variable fragments of SO(\(\exists\)) is examined.