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Preprint Number 614
614. Artem Chernikov and Saharon Shelah
On the number of Dedekind cuts and two-cardinal models of dependent theories
Submission date: 14 August 2013.
For an infinite cardinal k, let ded(k) denote the supremum of the number of Dedekind cuts in linear orders of size k. It is known that k < ded(k) \leq 2^k for all k, and that ded(k) < 2^k is consistent for any k of uncountable cofinality. We prove however that 2^k \leq ded(ded(ded(ded(k)))) 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.
Mathematics Subject Classification: 03E04, 03E10, 03E75, 03C45, 03C55
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