### Many countable support iterations of proper forcings preserve Souslin trees

by Mildenberger and Shelah. [MdSh:973]

Annals Pure and Applied Logic, 2014

We show that there are many models of {cov} M =
aleph_1 and cof {{M}} = aleph_2
in which the club
principle holds and there are Souslin trees. The proof
consists of the following main steps:
begin {enumerate}
item [1.] We give some iterable and some non-iterable
conditions on a forcing in terms of games that imply that the
forcing is (T,Y, {S})-preserving. A special case of
(T,Y, {S})-preserving is preserving the Souslinity of an
omega_1-tree.
item [2.]
We show that some tree-creature forcings from [RoSh:470]
satisfy the sufficient condition for one of the strongest games.
item [3.]
Without the games, we show that some linear creature forcings from
[RoSh:470] are (T,Y, {S})-preserving. There are non-Cohen
preserving examples.
item [4.] For the wider class of non-elementary proper forcings we
show that omega-Cohen preserving for certain candidates implies
(T,Y, {S})-preserving.
item [5.] (+ 1 978)We give a less general but hopefully more easily
readable
presentation of a result from [Sh:f, Chapter~18, section 3]: If all
iterands in a countable support iteration are proper and
(T,Y, {S})-preserving, then also the iteration is
(T,Y, {S})-preserving. This is a presentation of the
so-called case A in which a division in forcings that add
reals and those who do not is not needed.
end {enumerate}
In [Mi:clubdistr] we showed: Many proper forcings from
[RoSh:470] with finite or countable {{H}}(n) (see Section
2.1) force over a ground model with diamondsuit_{omega_1} in a
countable support iteration the club principle. After omega_1
iteration steps the diamond holds anyway.

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