Joint work with Ralf Schindler.
Abstract. We formulate combinatorial principles that combine the square principle with various strong forms of diamond, 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 $L$:
- For every infinite regular cardinal $\lambda$, there exists a special $\lambda^+$-Aronszajn tree whose projection is almost Souslin.
- For every infinite cardinal $\lambda$, there exists a respecting-$\lambda^+$-Kurepa tree. Roughly speaking, this means that this $\lambda^+$-Kurepa tree looks very much like a $\lambda^+$-Souslin tree.
A. Rinot and R. D. Schindler, Square with built-in diamond-plus, J. Symbolic Logic, 82(3): 809-833, 2017.