Abstract:

A classical theorem of Schur asserts that whenever the positive integers are finitely coloured, we can find x and y such that x, y and x+y are monochromatic. A moment's thought suffices to see that the same statement holds for the pattern x, y, xy. What about the common generalisation, the pattern x, y, x+y, xy? Although this question was posed no later than 1979, the answer was not known until recently even for the much simpler pattern x+y, xy. In the paper of Moreira which will be the basis of my talk, the author shows that for any finite colouring of the positive integers, there exists a monochromatic pattern x, x+y, xy. The proof, as befits this seminar, relies on dynamics.

A classical theorem of Schur asserts that whenever the positive integers are finitely coloured, we can find x and y such that x, y and x+y are monochromatic. A moment's thought suffices to see that the same statement holds for the pattern x, y, xy. What about the common generalisation, the pattern x, y, x+y, xy? Although this question was posed no later than 1979, the answer was not known until recently even for the much simpler pattern x+y, xy. In the paper of Moreira which will be the basis of my talk, the author shows that for any finite colouring of the positive integers, there exists a monochromatic pattern x, x+y, xy. The proof, as befits this seminar, relies on dynamics.

## Date:

Tue, 07/11/2017 - 12:00 to 13:00

## Location:

Manchester faculty lounge