Arithmetic Ramsey theory, graph theory and combinatorics

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In today's fast growing world of technology, many developments have been the result of old work done in pure mathematics. For example, many modern encryption techniques are based on number theory done in the 1600's, ideas that until recently seemed to be not much more than curious properties of the counting numbers. As another example, modern transportation or communication networks are presently analyzed and optimized using basic ideas from what is known as "graph theory", the study of properties of structures formed using dots and lines connecting the dots. My present research proposal considers the pure mathematics underlying many of these kind of applications, however I leave the "applying" to other specialists. I am trying to discover more of what I consider to be beautiful properties of graphs and number theory independent of how they are utilized. It is the symmetry and inherent "beauty" of both number theory and graph theory, and how they relate, that forms the bulk of my research. Another area I plan to study is similar to graph theory, called lattice theory, which studies special kinds of networks that have more "order" and symmetry.