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.