  For an infinite cardinal $\kappa$, let $ded\kappa$ denote the supremum of the
number of Dedekind cuts in linear orders of size $\kappa$. It is known that
$\kappa<ded\kappa\leq 2^{\kappa}$ for all $\kappa$ and that
$ded\kappa<2^{\kappa}$ is consistent for any $\kappa$ of uncountable
cofinality. We prove however that $2^{\kappa}\leq ded ( ded ( ded (
ded\kappa)))$ always holds. Using this result we calculate the Hanf numbers for
the existence of two-cardinal models with arbitrarily large gaps and for the
existence of arbitrarily large models omitting a type in the class of countable
dependent first-order theories. Specifically, we show that these bounds are as
large as in the class of all countable theories.
