Comparing the Real Line to Combinatorics of the Uncountable

Set theory is the area of mathematics that studies infinities. There are small infinities such as the size of the natural numbers 0, 1, 2, 3, , and there are larger infinities such as the collection of real numbers numbers that may be fractions or even irrational like pi. These infinities, which we call cardinals, increase without end leading to more and more complicated infinite sets. There are also other ways to think about small and large infinities rather than just brute size. For instance, we can think about events with large or small probability or special regions of a given space which are particularly dense or particularly spread out. Comparing these various ways of thinking about small and large quantities is the area of set theory known as cardinal characteristics. Our project places itself squarely within this field. In the project we will investigate how such notions of size play out on combinatorial structures such as linear orders. Thus our project has two parts to it the study of such structures on the one hand and the study of cardinal characteristics on the other hand. Despite being both important areas of modern set theory these two have not seen much interaction so our project has the potential to be innovative by bringing them together.