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Level: BSc, MSci, MSc
Title: Ramsey theory in the integers
Supervisor:

Dr Mark Walters
Research Area: Combinatorics
Description:

The starting point for this project would be Schur's theorem. This states that whenever the natural numbers are partitioned into k sets A_1, A_2, \ldots, A_k then we can solve the equation x + y = z with all 3 of x, y and z in the same one of the A_i. Rather than a partition we normally speak of colouring the natural numbers with k colours where the numbers in A_1 get the first colour, the numbers in A_2 get the second colour and so on. Then Schur's theorem states that we can find a monochromatic solution (i.e. a solution where all numbers have the same colour) to the equation x + y = z.

There are many results of the same type which vary from Schur's theorem above (an easy consequence of Ramsey's theorem) to very deep intricate results and, indeed, many open questions. Thus this project is suitable for all levels: third year, MSci and MSc as there is a lot of choice about which material to cover.
Further Reading:
  • R. L. Graham, B. L. Rothschild, J. H. Spencer, Ramsey Theory. Second edition. Wiley, Chichester, 1990.
Key Modules:

MTH6109 Combinatorics 
Other Information:
Current Availability: Yes