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

by Chernikov and Shelah. [CeSh:1035]

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 <= 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} <= 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.

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