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

by Chernikov and Shelah. [CeSh:1035]

J Institute of Math of Jussieu, 2016

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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