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 $\kappa$.