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For an infinite cardinal $\kappa$, let $ded\kappa$ denote the supremum of thenumber 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 uncountablecofinality. We prove however that $2^{\kappa}\leq ded ( ded ( ded (ded\kappa)))$ always holds. Using this result we calculate the Hanf numbers forthe existence of two-cardinal models with arbitrarily large gaps and for theexistence of arbitrarily large models omitting a type in the class of countabledependent first-order theories. Specifically, we show that these bounds are aslarge as in the class of all countable theories.

On the number of Dedekind cuts and two-cardinal models of dependent theories