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.
