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.