We write κ → (σ : τ)γ2 for: Every partition f : [κ]2 → γ has a homogeneous set
[A;B] := {{α,β}| α 
A,β B} where α < β for all α A and β B and card(A) = σ,
card(B) = τ, i.e. f is constant on [A;B].
We write κ ⁄→ (σ : τ)γ2 for the negation of this statement.
Theorem (see above)
If there exitsts an (ω1,1)-morass, then there is a ccc forcing for ω2 ⁄→ (ω : 2)ω2.
Question
Assume that there is an (ω1,2)-morass. Does then exist a ccc forcing with finite conditions
which forces ω3 ⁄→ (ω : 2)ω12?
Question (Todorcevic)
Can there exist (consistently) a function f : ω3 × ω3 → ω such that f is not constant on any
rectangle A × B with infinite A,B ⊆ ω3?
Question (famous)
Is the existence of an (ω,ω3)-superatomic or an (ω1,ω3)-superatomic Boolean algebra
consistent?
Even though the method generalizes straightforwardly to higher-dimensions, this is not true
for the consistency statements. The reason is that the conditions of the forcing have to fit
together in more directions, if we go to higher dimensions.
Question Can we find good applications in dimensions higher than two (three)?
Even though the method is inspired by iterated forcing, all my examples use basically finite
sets of ordinals as conditions. Hence names are not really needed.
Question Can we find an application which uses names?
Question Can we do something similar for countable support?
Motivation
Two-dimensional applications
Three-dimensional application
Morasses / Example