On Winning Strategies with Unary Quantifiers

A combinatorial argument for two finite structures to agree on all sentences with bounded quantifier rank in first-order logic with any set of unary generalized quantifiers is given. It is known that connectivity of finite structures is neither in monadic Σ¹1 nor in ≲ww (Qu), where Qu is the set of all unary generalized quantifiers. Using this combinatorial argument and a combination of second-order Ehrenfeucht-Fraïssé games developed by Ajtai and Fagin, we prove that connectivity of finite structures is not in monadic σ¹1 with any set of unary quantifiers even if sentences are allowed to contain built-in relations of moderate degree. The combinatorial argument is also used to show that no class (if it is not in some sense trivial) of finite graphs defined by forbidden minors is in ≲ww(Qu). In particular, the class of planar graphs is not in ≲ww(Qu)