Monadic second order finite satisfiability and unbounded tree-width

The finite satisfiability problem of monadic second order logic is decidable only on classes of structures of bounded tree-width by the classic result of Seese (1991). We prove the following problem is decidable: Input: (i) A monadic second order logic sentence $alpha$, and (ii) a sentence $beta$ in the two-variable fragment of first order logic extended with counting quantifiers. The vocabularies of $alpha$ and $beta$ may intersect. Output: Is there a finite structure which satisfies $alphalandbeta$ such that the restriction of the structure to the vocabulary of $alpha$ has bounded tree-width? (The tree-width of the desired structure is not bounded.) As a consequence, we prove the decidability of the satisfiability problem by a finite structure of bounded tree-width of a logic extending monadic second order logic with linear cardinality constraints of the form $|X_{1}|+cdots+|X_{r}|<|Y_{1}|+cdots+|Y_{s}|$, where the $X_{i}$ and $Y_{j}$ are monadic second order variables. We prove the decidability of a similar extension of WS1S.