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.

**Downloads:**

Submitted to Journal of Symbolic Logic, October 2015.

Accepted, November 2016.

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