Abstract

Several classical results in Ramsey theory (including famous theorems of Schur, van der Waerden, Rado) deal with finding monochromatic linear patterns in two-colourings of the integers. Our topic will be quantitative extensions of such results. A linear system L over F_q is common if the number of monochromatic solutions to L=0 in any two-colouring of F_qⁿ is asymptotically at least the expected number of monochromatic solutions in a random two-colouring of F_qⁿ. Motivated by existing results for specific systems (such as Schur triples and arithmetic progressions), as well as extensive research on common and Sidorenko graphs, the systematic study of common systems of linear equations was recently initiated by Saad and Wolf. Fox, Pham and Zhao characterised common linear equations. I will talk about recent progress towards a classification of common systems of two or more linear equations. In particular, the uncommonness of an arbitrarily large system L can be reduced to studying single equations implied by L. Joint work with Anita Liebenau and Natasha Morrison.