  lambda-good frame is for us a parallel of the class of models of a
superstable theory. Our main line is to start with lambda-good^+ frame s,
categorical in lambda, n-successful for n large enough and try to have parallel
of stability theory for K_{s(+l)} for l<n not too large. Characteristically
from time to time we have to increase n relative to l to get our desirable
properties; we do not critically mind the exact n, so one may think of an
omega-successful s. A posteriori we are interested in the model theory of such
classes K_s per-se, and see as a test for this theory, that in the
omega-successful case we can understand also the model in higher cardinals,
e.g., prove that K^s_mu is not empty for every mu>=lambda. Recall there are
reasonable lambda-frames which are not n-excellent but still we can say alot on
models in $K_{s(+l)} for l<n. Moving from lambda to lambda^+ we would have
preferred not to restrict ourselves to saturated models but at present we do
not know it. However, in the omega-excellent case we can understand the class
of lambda^{+omega}--saturated models in $K_s$, i.e., K^{s(+omega)}. This fits
well the thesis that it is reasonable to first analyze the quite saturated
case.
