# SQUARE with BUILT-IN DIAMOND-PLUS

Assaf Rinot, Ralf Schindler

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

## 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 means that this λ+-Kurepa tree looks very much like the λ+-Souslin trees that Jensen constructed in L.

Original language English 809-833 25 Journal of Symbolic Logic 82 3 https://doi.org/10.1017/jsl.2016.69 Published - 1 Sep 2017

### Funding

Finally, let α ∈ §, and we shall show that Tα ⊆ Im(bα). Let y ∈ Tα be arbitrary. By definitionofT,wehavey∈Nα.SinceNαisrud-closed,theset{γ<α|y(γ)=1} is in Nα, and so by α ∈ §, there exists some β ∈ Cα such that fα(β) = {γ < α | y(γ) = 1}. Let β′ = α−1(β). Then gα(β′) = y. Now, put x := y β′. Then x ∈ T Dα, and by definition of bα, we have bα(x) = y, as sought. ⊣ Corollary 4.16. Suppose that ♦ † holds for a given uncountable cardinal . Then thereexistsa (E)-respecting++-Kurepatreethathasno+-Aronszajnsubtrees. cf( ) Proof. The construction of all involved objects is identical to that of the proof of Theorem 4.15, but this time we consult with a ♦†-sequence rather than ♦+. Consequently, the reflection argument of [5, Theorem 2] shows that the +-Kurepa tree will contain no +-Aronszajn subtrees. ⊣ §5. Acknowledgements. This work was engaged when the authors met at the MAMLS meeting at Carnegie Mellon University, May 2015. We are grateful to the organizers for the invitation. The first author was partially supported by grant No. 1630/14 of the Israel Science Foundation. He would also like to acknowledge the German-Israeli Foundation for Scientific Research and Development, grant No. I-2354-304.6/2014, for supporting his travel to the second author on July 2015. The second author partially supported by the SFB 878 of the Deutsche Forschungsgemeinschaft (DFG). The authors thank the referee for his feedback.

FundersFunder number
SFB
Carnegie Mellon University1630/14
Deutsche Forschungsgemeinschaft
German-Israeli Foundation for Scientific Research and DevelopmentI-2354-304.6/2014, .6/2014, I-2354-304
Israel Science Foundation

## Keywords

• Kurepa tree
• almost Souslin tree
• constructibility
• diamond principle
• parameterized proxy principle
• square principle
• walks on ordinals

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