# Square with built-in diamond-plus

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 $L$:

1. For every infinite regular cardinal $\lambda$, there exists a special $\lambda^+$-Aronszajn tree whose projection is almost Souslin.
2. 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.