Theorem
If there exists a simplified (ω1,1)-morass, then there is a ccc forcing of size ω1 that adds an
ω2-Suslin tree.
It was known before that there exists a ccc forcing which adds an ω2-Suslin tree if □ω1 holds
(Todorcevic).
Theorem
Assume that there exists a (simplified) (ω1,1)-morass. Then there is a ccc forcing which adds
a g : [ω2]2 → ω such that {ξ < α | g(ξ,α) = g(ξ,β)} is finite for all α < β < ω2.
This was first proved by Todorcevic using only the assumption that □ω1 holds. He uses ordinal
walks / Δ-functions.
Note that, by the Erdös-Rado theorem, the existence of a function g like in the theorem
implies the negation of CH.
Theorem
There exists consistently a chain < Xα | α < ω2 > such that Xα ⊆ ω1, Xβ -Xα is finite and
Xα - Xβ has size ω1 for all β < α < ω2.
This was first proved by Koszmider using ordinal walks.
Theorem
There exists consistently a family < fα | α < ω2 > of functions such that {ξ < ω1 | fα(ξ) = fβ(ξ)}
is finite for all α < β < ω2.
This was first proved by Zapletal in a more general form.
Theorem
There exists consistently an (ω,ω2)-superatomic Boolean algebra.
This was first proved by Baumgartner and Shelah independently of ordinal walks. Later
Todorcevic found in the presence of □ω1 a ccc forcing which proves the consistency.
Motivation
Three-dimensional application
Open Problems
Morasses / Example