On cuts in ultraproducts of linear orders I

Abstract

For an ultrafilter $D$ on a cardinal $\kappa,$ we wonder for which pair $(\theta_1, \theta_2)$ of regular cardinals, we have: for any $(\theta_1+\theta_2)^+-$saturated dense linear order $J, J^{\kappa}/ D$ has a cut of cofinality $(\theta_1, \theta_2).$ We deal mainly with the case $\theta_1, \theta_2 > 2^\kappa.$