SQUARE with BUILT-IN DIAMOND-PLUS

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.