On some downward transfer properties in Foreman-Laver model (Infinite Combinatorics and Forcing Theory)

Abstract

In [FL88], Foreman and Laver proved, assuming the existence of a huge cardinal the consistency of the transfer property of maximal chromatic number from aleph_{2} to aleph_{1}. We combine this result with another result by Baumgartner [Bau84] to prove, also assuming the existence of a huge cardinal, that such a transfer property from aleph_{2} to aleph_{1} does not follow from the transfer property from aleph_{3} to aleph_{1}.