A weak embedding between trees is a function that preserves the strict order. A class \(U\) of trees is said to be universal for a class \(C\) of trees if every tree in \(C\) weakly embeds in an element of \(U\). It turns out that the pre-ordered structure induced by weak embeddability on a class \(C\) of trees is a plausible tool for the study of the elements of \(C\). One can ask e.g., what is the universality number of a class of trees (the size of the smallest subclass which is universal)? can it be 1? whether a subclass is cofinal? etc. If CH holds, then the class of \(\aleph_1\)-wide Aronszan trees (trees of height and size \(\aleph_1\) without cofinal branches) does not have a maximal tree under weak embeddability (this follows from Kurepa's works). Todorcevic has proved, among other things, that under MA\(_{\aleph_1}\), the class of Aronszajn trees has no maximal object. In their joint work on wide Aronszajn trees under MA\(_{\aleph_1}\), Dzamonja and Shelah introduced the notion of a specializing triple that connects weak embeddings to the specialization of trees. In particular, they reproved Todorcevic's result using specializing triples. In this talk, we shall focus on a variant of this notion in a general setting and demonstrate the main aspects of it. We shall then discuss some negative results on the universality problem for Aronszajn trees whose height is the successor of a regular cardinal, and hopefully, we shall finish the talk with some open problems.

The results have been obtained in a collaboration with Mirna Dzamonja.