  We introduce the notion of a w-good $\lambda$-frame which is a weakening of
Shelah's notion of a good $\lambda$-frame. Existence of a w-good
$\lambda$-frame implies existence of a model of size $\lambda^{++}$. Tameness
and amalgamation imply extension of a w-good $\lambda$-frame to larger models.
As an application we show:
  $\textbf{Theorem}$ Suppose $2^{\lambda}< 2^{\lambda^{+}} < 2^{\lambda^{++}}$
and $2^{\lambda^{+}} > \lambda^{++}$. If $I(K, \lambda) = I(K, \lambda^{+}) = 1
\leq I(K, \lambda^{++}) < 2^{\lambda^{++}}$ and $K$ is $(\lambda,
\lambda^+)$-tame, then $K_{\lambda^{+++}} \neq \emptyset$.
  The proof presented clarifies some of the details of the main theorem of
[Sh576] and avoids using the heavy set-theoretic machinery of [Sh: h \S VII] by
replacing it with tameness.
