A topological generalization of partition regularity

Abstract

In 1939, Richard Rado showed that any complex matrix is partition regular over $ℂ$ if and only if it satisfies the columns condition. Recently, Hogben and McLeod explored the linear algebraic properties of matrices satisfying partition regularity. We further the discourse by generalizing the notion of partition regularity beyond systems of linear equations to topological surfaces and graphs. We begin by defining, for an arbitrary matrix $\Phi$, the metric space ($M_{\Phi }$, $\delta$). Here, $M_{\Phi }$ is the set of all matrices equivalent to $\Phi$ that are (not) partition regular if $\Phi$ is (not) partition regular; and for elementary matrices, $E_i$ and $F_j$, we let $\delta \left(A,B\right)=min\left\{m=l+k:B=E_1\dots E_{l}A{F}_1\dots F_k\right\}$. Subsequently, we illustrate that partition regularity is in fact a local property in the topological sense, and uncover some of the properties of partition regularity from this perspective. We then use these properties to establish that all compact topological surfaces are partition regular.