SQUARE WITH BUILT-IN DIAMOND-PLUS

@article{Rinot2017SQUAREWB,
  title={SQUARE WITH BUILT-IN DIAMOND-PLUS},
  author={Assaf Rinot and Ralf Schindler},
  journal={The Journal of Symbolic Logic},
  year={2017},
  volume={82},
  pages={809 - 833}
}
Abstract We formulate combinatorial principles that combine the square principle with various strong forms of the diamond principle, and prove that the strongest amongst them holds in L for every infinite cardinal. As an application, we prove that the following two hold in L: 1. For every infinite regular cardinal λ, there exists a special λ+-Aronszajn tree whose projection is almost Souslin; 2. For every infinite cardinal λ, there exists a respecting λ+-Kurepa tree; Roughly speaking, this… 
Distributive Aronszajn trees
Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ entails the existence of a normal $\lambda$-distributive
Knaster and friends I: closed colorings and precalibers
The productivity of the $$\kappa $$κ-chain condition, where $$\kappa $$κ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent
REDUCED POWERS OF SOUSLIN TREES
We study the relationship between a $\unicode[STIX]{x1D705}$ -Souslin tree $T$ and its reduced powers $T^{\unicode[STIX]{x1D703}}/{\mathcal{U}}$ . Previous works addressed this problem from the

References

SHOWING 1-10 OF 25 REFERENCES
Stacking mice
TLDR
The main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over Kc∥κ, and it is shown that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a properclass of Woodin cardinals.
Chain conditions of Products, and Weakly Compact Cardinals
TLDR
It is proved that for every regular cardinal $\kappa > \aleph _1 {\rm{,}}$ the principle □( k ) is equivalent to the existence of a certain strong coloring $c\,:\,[k]^2 \, \to $ k for which the family of fibers ${\cal T}\left( c \right)$ is a nonspecial κ -Aronszajn tree.
SOME REMARKS ON NON-SPECIAL COHERENT ARONSZAJN TREES
We introduce some guessing principles sufficient for the existence of non-special coherent Aronszajn trees and show how they relate to some of the standard principles in Set Theory (like MAω1 and ♦).
A microscopic approach to Souslin-tree constructions, Part I
The uniformization property for ℵ2
AbstractWe present S. Shelah’s result thatS12={δ<ω2: cf(δ)=ω1} may have the uniformization property (cf., §1, or [3] for a definition) for “well-chosen sequences”, 〈ηδ:δ∈S12^ηδ an
Representing trees as relatively compact subsets of the first Baire class
We show that there is a scattered compact subset K of the first Baire class a Baire space X and a separately continuous mapping f: X × K → R which is not continuous on any set of the form G × K,
Hedetniemi's conjecture for uncountable graphs
It is proved that in Godel's constructible universe, for every infinite successor cardinal k, there exist graphs G and H of size and chromatic number k, for which the tensor product graph (G x H) is
Club guessing sequences and filters
TLDR
It is proved that for every uncountable regular cardinal μ, relative to the existence of a Woodin cardinal above μ, it is consistent that every tail club guessing ideal on μ is precipitous.
Some results about (+) proved by iterated forcing
TLDR
The consistency of CH+⌝(+) and CH+(+)+there are no club guessing sequences on ω 1 is shown and it is proved that ◊+ does not imply the existence of a strong club guessing sequence on ϊ 1.
Putting a diamond inside the square
By a 35‐year‐old theorem of Shelah, □λ+♦(λ+) does not imply for regular uncountable cardinals λ .
...
...