When a set theorist hears "combinatorics", Part 2: Trees

Abstract

In the previous installment of this series (When a Set Theorist Hears “Combinatorics”: Ramsey Theory, [Gil25]) we discussed the combinatorial problem of finding large patches of order in graphs, and this was done from the perspective of a set theorist. Thus we focused on infinite graphs, both countably infinite and uncountably infinite. In this follow-up expository article, we focus on trees, another common object in combinatorics. Working yet again from the perspective of a set theorist, we will focus on infinite trees. We shall begin with graph-theoretic trees to set the stage. After subsequently developing the tools to talk about trees with a “longer than infinite” height, we will consider several kinds of uncountable trees, how they structurally differ from countably infinite trees, and how they are related to a question about characterizing the real numbers.