  One of Courcelle's celebrated results states that if C is a class of graphs
of bounded tree-width, then model-checking for monadic second order logic is
fixed-parameter tractable on C by linear time parameterised algorithms. An
immediate question is whether this is best possible or whether the result can
be extended to classes of unbounded tree-width.
  In this paper we show that in terms of tree-width, the theorem can not be
extended much further. More specifically, we show that if C is a class of
graphs which is closed under colourings and satisfies certain constructibility
conditions such that the tree-width of C is not bounded by log^{16}(n) then
MSO_2-model checking is not fixed-parameter tractable unless the satisfiability
problem SAT for propositional logic can be solved in sub-exponential time. If
the tree-width of C is not poly-logarithmically bounded, then MSO_2-model
checking is not fixed-parameter tractable unless all problems in the
polynomial-time hierarchy, and hence in particular all problems in NP, can be
solved in sub-exponential time.
