Latin Subrectangles

In the study of mathematical structures many fundamental questions revolve around existence or absence of substructures of the same type as the parent object. This thesis answers several prominent and longstanding such questions for a structure known as a Latin rectangle. These structures have been studied for over three centuries, and have a wealth of applications in practical areas ranging from the design of statistical experiments to codes for data storage and communication. We now have a greater understanding of how many subrectangles Latin rectangles are likely to possess, and also how to construct Latin rectangles that have no subrectangles.