  Let $X$ be a Banach space and $K$ an absolutely convex, weak$^\ast$-compact
subset of $X^\ast$. We study consequences of $K$ having a large or undefined
Szlenk index and subsequently derive a number of related results concerning
basic sequences and universal operators. We show that if $X$ has a countable
Szlenk index then $X$ admits a subspace $Y$ such that $Y$ has a basis and the
Szlenk indices of $Y$ are comparable to the Szlenk indices of $X$. If $X$ is
separable, then $X$ also admits subspace $Z$ such that the quotient $X/Z$ has a
basis and the Szlenk indices of $X/Z$ are comparable to the Szlenk indices of
$X$. We also show that for a given ordinal $\xi$ the class of operators whose
Szlenk index is not an ordinal less than or equal to $\xi$ admits a universal
element if and only if $\xi<\omega_1$; W.B. Johnson's theorem that the formal
identity map from $\ell_1$ to $\ell_\infty$ is a universal non-compact operator
is then obtained as a corollary. Stronger results are obtained for operators
having separable codomain.
