An Overview of Davies Trees, with Applications

A frequent strategy in infinitary combinatorics is to enumerate the desired objectives and meet them inductively. The key to making this strategy work is to find the correct enumeration. Increasing chains of elementary submodels are particularly powerful ways to proceed, since they allow the construction of increasing filtrations of the objectives which reflect many of the properties of the global structure to the local situations in which we are meeting the objectives. While this technique is quite useful in a number of circumstances,  the requirement that the filtrations be increasing places, at times, substantial constraints on how many objectives one may meet.

In this talk, we introduce the idea of Davies trees. These are trees of elementary submodels (rather than chains) which allow the construction of more complicated - and more widely applicable - filtrations of objectives. In particular, we will define Davies trees and provide examples of how to use them.