Rado's Selection Principle: Equivalences and Applications

Abstract

Rado's Selection Principle is a combinatorial theorem which allows the characterization of infinite objects (e.g. graphs, groups, partially-ordered sets) based on the characterization of their finite subparts. That is, a typical result of the application of Rado's Selection Principle would be a theorem of the following sort: Object A has property P if and only if every finite subobject of A has property P. The necessity of the second hypothesis is usually obvious because the subobjects usually inherit the properties of the objects (in workable applications), so Rado is used to prove sufficiency. Theorems of this sort are extremely useful because it is normally possible to check directly a condition on a finite object, and impossible to do so on an infinite one. This description is, of course, far too general, but it gives some indication of types of problems here undertaken. [From Introduction]