Title: Some groups which are extraordinarily close to being free
Abstract: Almost free groups (groups whose subgroups of smaller cardinality are free) have been a subject of study over the last 60 years. Some such groups can be made to satisfy further conditions which are satisfied by free groups, while still failing to be free. Combinatorial set theory often plays a role in such constructions. I'll give a review of almost free groups, including history and some examples, as well as present a very recent result.