In this talk we will continue our study of 2-cardinals
(versions of Velleman's simplified morasses). We will use them to define
a certain $\kappa$-valued coloring on pairs of $\kappa+$ ($\kappa$ is a cardinal) and a
family of functions from $\kappa+$ to $\{0,1,2\}$ with interesting combinatorial
properties, which we will use to construct a $\kappa$-Kurepa tree and a version
of the Hausdorff gap at the level of $\kappa$.