One of the driving principles behind the mathematical branch of Ramsey Theory is that in any large system there are small substructures inside that system that are ordered. While we expect this behavior for any ``large'' system and any reasonable definition of ``order'', we are often interested in the exact relationship between the size of the system and the ordered substructures that must exist. In this project, students will explore open questions in this area, where the large system is an edgecolored complete graph and the ordered substructures are small fixed graphs with certain color patterns. In particular, we will consider problems such as determining the least number of colors we can use to color the edges of a complete graph on n vertices such that there do not exist two colorisomorphic triangles in the coloring, and many others.
Graph theory is required, combinatorics is preferred.