Graph Theory and Combinatorics

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The underlying motto of Ramsey theory is that total disorder is impossible. Ramsey-type theorems also have roots in different branches of mathematics, and the theory developed from them has influenced diverse areas such as number theory, logic, probability theory, geometry and theoretical computer science. Ramsey theory for graphs has a long and rich history with applications in social study, infrastructure, telecommunication and biological networks. This area of research has resulted in development of powerful techniques, such as Szemeredi's regularity lemma and dependent random choice. However, much less is known for Ramsey theory for hypergraphs. For instance the Ramsey number of k-uniform cycles are not known, whereas its graph version is known since the 1970s. The focus of the project is to study question in this area. Another topic of interest will be to classify sparse hypergraphs that has linear Ramsey number.