Sh:1035

• Chernikov, A., & Shelah, S. (2016). On the number of Dedekind cuts and two-cardinal models of dependent theories. J. Inst. Math. Jussieu, 15(4), 771–784. arXiv: 1308.3099 DOI: 10.1017/S1474748015000018 MR: 3569076
• Abstract:
  For an infinite cardinal \kappa, let {\rm ded}\kappa denote the supremum of the number of Dedekind cuts in linear orders of size \kappa. It is known that \kappa<{\rm ded}\kappa \leq2^{\kappa} for all \kappa and that {\rm ded}\kappa< 2^{\kappa} is consistent for any \kappa of uncountable cofinality. We prove however that 2^{\kappa}\leq{\rm ded}\left( {\rm ded}\left({\rm ded}\left({\rm ded} \kappa\right)\right)\right) 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.
• Version 2014-12-30_11 (15p) published version (14p)

@article{Sh:1035,
 author = {Chernikov, Artem and Shelah, Saharon},
 title = {{On the number of Dedekind cuts and two-cardinal models of dependent theories}},
 journal = {J. Inst. Math. Jussieu},
 fjournal = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Math\'ematiques de Jussieu},
 volume = {15},
 number = {4},
 year = {2016},
 pages = {771--784},
 issn = {1474-7480},
 mrnumber = {3569076},
 mrclass = {03C45 (03C55 03E04 06A05)},
 doi = {10.1017/S1474748015000018},
 note = {\href{https://arxiv.org/abs/1308.3099}{arXiv: 1308.3099}},
 arxiv_number = {1308.3099}
}