Research Fellows Directory
Dr John Talbot
University College London
My research is in a branch of pure mathematics called extremal combinatorics. As a branch of pure mathematics this is relatively young - the first proper results in this area are only a hundred years old!
Very generally, extremal combinatorics deals with questions of how large or dense a combinatorial structure can be if it is to possess a given property. For example in any network in which at least half of all possible pairs of nodes are connected there must be a triangle. Another well known example is that amongst any group of six people there are always three mutual friends or three mutual strangers. The first example is a Turan result - it says that a given density of connections in a network implies a property of the network (i.e. it must contain a triangle). The second example is a Ramsey result - it says that in any large group of people there will always be certain subgroups with a special property (i.e. three friends or three strangers).
My own research aims to answer the following question: given a hypergraph (this is an abstract generalisation of a network) in which a fixed proportion of the connections are present when must it contain a part that has significantly more connections?
Applications of combinatorics are widespread. Whenever we analyse networks - such as the Internet our understanding of the mathematics of combinatorics can help us to identify interesting features. However my own research is not in general guided by applications, rather I aim to answer the central outstanding questions of pure mathematical interest in this area. By hopefully contributing new results and ideas to pure mathematics there are more tools at the disposal of future generations of applied researchers.