Ramsey Theory: Avoiding Trees with Few Colours




Lane, Andrew

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This research project concerns an area of mathematics called graph theory. Graph theory studies the structure of networks called graphs. The nodes of a graph are called vertices, and the connections are called edges. Many problems concern graph colouring, where the vertices or edges are each assigned a value. In particular, this project studies a problem in Ramsey theory. Ramsey theory tells us that in any large enough network, there must be structured parts, and we are particularly interested in how large the network must be to contain such structured parts. The function f(n,H,q) is the minimum number of colours needed to colour the edges of the complete graph Kn, which contains all possible edges, so that every copy of the graph H within the complete graph is coloured with at least q colours. The behavior of f(n,H,q) when H is a complete graph or path has been studied by Erdős, Gyárfás, Krueger, and many others. This project focuses on the case in which the graph H is a tree. It investigates numerical bounds on f(n,H,q) for small trees and asymptotic bounds for larger trees, fully characterizing the asymptotic bounds on f(n,H,q) for a particular class of trees.



mathematics, graph theory, Ramsey theory, combinatorics, graph colouring