Uncommon Systems of Equations

EasyChair Preprint no. 5596

Abstract

A system of linear equations L over a finite field F is common if the number of monochromatic solutions to L in any two-colouring of F^n is asymptotically at least the number of monochromatic solutions in a random two-colouring of F^n. The line of research on common systems of linear equations was recently initiated by Saad and Wolf. They were motivated by existing results for specific systems (such as Schur triples and arithmetic progressions), as well as extensive research on common and Sidorenko graphs. Building on earlier work, Fox, Pham and Zhao characterised common linear equations. For systems of two or more equations, only sporadic results were known.

We prove that any system containing an arithmetic progression of length four is uncommon, confirming a conjecture of Saad and Wolf. This follows from a stronger result which allows us to deduce the uncommonness of a general system from considering certain one- or two-equation subsystems.

Keyphrases: common graph, Fourier analysis, linear system, Ramsey theory

@Booklet{EasyChair:5596,
  author = {Nina Kamčev and Anita Liebenau and Natasha Morrison},
  title = {Uncommon Systems of Equations},
  howpublished = {EasyChair Preprint no. 5596},

  year = {EasyChair, 2021}}