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.
