  We study a new extension of the weak MSO logic, talking about boundedness.
Instead of a previously considered quantifier U, expressing the fact that there
exist arbitrarily large finite sets satisfying a given property, we consider a
generalized quantifier U, expressing the fact that there exist tuples of
arbitrarily large finite sets satisfying a given property. First, we prove that
the new logic WMSO+U_tup is strictly more expressive than WMSO+U. In
particular, WMSO+U_tup is able to express the so-called simultaneous
unboundedness property, for which we prove that it is not expressible in
WMSO+U. Second, we prove that it is decidable whether the tree generated by a
given higher-order recursion scheme satisfies a given sentence of WMSO+K_tup.
