Ramsey theory is, loosely, the study of how structure can
arise in disordered mathematical systems. The classical Ramsey theory
is a generalisation of the observation that at a party, if you choose
six people at random, then either three people will know each other or
three will be mutual strangers. A 3rd year project might look at
graph theory, at Ramsey's theorem and its immediate generalisation, at
the Van der Waerden and Hales-Jewitt theorems. Further directions
would be to look at the elegant dynamical systems proofs, and/or to
investigate the more recent work of Gowers, Green and Tao. 4th year
students might also look at the measure-theoretic issues around
Szemeredi's theorem and Hindman's theorem, and Furstenburg's proof of
them.
Books
Ramsey theory, Ronald L. Graham, Bruce L. Rothschild, Joel H. Spencer.
Ramsey theory on the integers, Bruce M. Landman, Aaron Robertson.
Recurrence in ergodic theory and combinatorial number theory, H.
Furstenberg.
Ergodic theory, Karl Petersen.