Universality problem for wide Aronszajn trees, part 1

The university problem for a class K of structures asks if there is a structure M in K such that every other structure in K embeds into. I will start by relating the universality problem for the class of wide Aronszajn trees to generalized descriptive set theory, and specifically to generalizations of Scott's theorem on the complexity of isomorphism classes in generalized Baire spaces. Next, we will see that in L there is a strong failure of this universality problem. Finally, I will describe a consistency result showing that a universal wide Aronszajn tree on omega_2 can exist relative to a weakly compact cardinal.  The new results are from a joint work with Menachem Magidor and Jouko Vaananen.