 Almost disjoint family on a regular cardinal

A family of size-$\kappa$ subsets of a set is almost disjoint at $\kappa$ when distinct members have intersection of size less than $\kappa$. Graphs of a long chain under eventual domination give such a family of size ${\kappa }^{+}$ on $\kappa \times \kappa$. Transport by a bijection gives a family on $\kappa$ itself.