  It is shown that every normalized weakly null sequence of length
$\kappa_{\lambda}$ in a Banach space has a subsequence of length $\lambda$
which is an unconditional basic sequence; here $\kappa_{\lambda}$ is a large
cardinal depending on a given infinite cardinal $\lambda$. Transfinite
topological games on Banach spaces are analyzed which determine the existence
of a long unconditional basic sequence. Then 'asymptotic disentanglement'
condition in a transfinite setting is studied which ensures a winning strategy
for the unconditional basic sequence builder in the above game. The following
problem is investigated: When does a Markushevich basic sequence with length
uncountable regular cardinal $\kappa$ admit a subsequence of the same length
which is a bimonotone basic sequence? Stabilizations of projectional
resolutions of the identity (PRI) are performed under a density contravariance
principle to gain some additional strong regularity properties, such as
bimonotonicity.
