This is the last of the series of talks devoted to present
applications of 2-cardinals (versions of Velleman's neat simplified
morasses) to combinatorics. We will use 2-cardinals to prove the
existence of a version of the Hausdorff gap at level $\kappa$ ($\kappa$ is a regular
cardinal) and use the coloring defined from 2-cardinals (a version of
Todorcevic's $\rho$ function) to prove the existence of a $\kappa_+$-Aronszajn tree.