Ramsey-Theoretic Results on Matrices Over a Finite Field

This thesis explores the unavoidable substructures of very large matrices with entries over a finite field. Several outcomes also apply to matrices with entries over a finite set or a finite set with a special element called 0. Our main theorems require some preparation to describe and are beyond the scope of an abstract. In essence, the first main theorem tells us that every sufficiently large matrix has a large and highly structured permuted submatrix in one of three specific ways. The second main theorem uses the first main theorem to show that, up to row-scaling, every sufficiently large matrix has a large permuted submatrix that is highly structured in one of nine specific ways. On the way to proving our second main theorem, we prove several other Ramsey-theoretic results for matrices. This research is motivated by Jim Geelen, Bert Gerards, and Geoff Whittle's need for such results in their own Ramsey-Theoretic work on matroids.