Number Theory Days in Lille

Bootstrapping partition regularity of linear systems

8 juil. 2019, 15:30

1h

Salle de réunion (Université de Lille)

Analytic-additive number theory

Orateur

Tom Sanders (University of Oxford)

Description

Suppose that $A$ is a $k\times d$ matrix of integers such that fir any $r$ there is some $N$ such that any $r$-colouring of $\{1,\dots ,N\}$ contains a monochromatic solution to $A$, meaning there is a colour class $C$ and $x\in C^d$ such that $Ax=0$. Not all matrices $A$ have this property (consider, for example, when all the entries of $A$ are positive), but when they do they are called partition regular. In this talk we consider what bounds can be given on $N$ in terms of $r$ (and $A$) when $A$ is partition regular.