# How fast is the continuum changing with respect to the relative change of size of the forcing notion?

Has there been any research done on the related rates of forcing?

If I force to increase the size of the continuum $$\mathfrak{c}$$ by 5 $$\aleph$$'s, say from $$\aleph_2$$ to $$\aleph_7$$, how fast does the size of the notion of forcing $$\mathbb{P}$$ change from the ground model to the forcing extension?

It seems they must just have a constant ratio, but perhaps there are small forcings which have a very big effect.

Gitik an I have results, which simply say that (sometimes under the assumption of the existence of large cardinals) one can have a pair (W, V) of models of ZFC, such that adding an $$Add(\omega, \kappa)$$-generic over V, adds an $$Add(\omega, \lambda)$$-generic over W, for some $$\lambda > \kappa.$$