Title: On winning strategies with unary quantifiers
Author: Juha Nurmonen

Abstract:
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 $\Sigma_1^1$ 
nor in $\ll_{\omega\omega}({\bf Q}_u)$, where ${\bf Q}_u$ is the set of
all unary generalized quantifiers. Using this combinatorial argument and 
a combination of second-order Ehrenfeucht-\Fraisse games developed by
Ajtai and Fagin, we prove that connectivity of finite structures is not 
in monadic $\Sigma_1^1$ 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 $\ll_{\omega\omega}({\bf Q}_u)$. Especially, the class of planar
graphs is not in $\ll_{\omega\omega}({\bf Q}_u)$.