Extremal graph theory, Ramsey theory and additive combinatorics

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In the first year of my PhD I was working on three topics in combinatorics (also known as discrete mathematics). One of them is extremal graph theory which is concerned with determining the maximal number of edges in a graph which does not contain a given subgraph. The second one is Ramsey theory. Roughly speaking, this field focuses on finding structures in a graph or its complement. Finally, the third area is additive combinatorics, which investigates subsets of groups. (Groups are objects on which addition can be defined, such as the integers or real numbers.) This last project also has significance in theoretical computer science.Together with my supervisor, Timothy Gowers, we have managed to establish publishable results in all three of these areas. A common technique which was used heavily in these projects is the so-called probabilistic method, which is a powerful tool allowing us to find objects with certain properties by randomly picking one object of all the available ones.In the future I will be aiming to solve further open problems in the three areas mentioned above.