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


For an infinite cardinal κ, let ded κ denote the supremum of the number of Dedekind cuts in linear orders of size κ. It is known that κ < ded κ ≤2κ for all κ and that ded κ < 2κ is consistent for any κ of uncountable cofinality. We prove however that 2κ≤ ded(ded(ded(ded κ))) 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.

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