#### Vol. 3, No. 4, 2010

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A topological generalization of partition regularity

### Liam Solus

Vol. 3 (2010), No. 4, 421–433
##### 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.

##### Keywords
partition regularity, columns condition, graphs, metric space, discrete topology, topological surface, triangulation
##### Mathematical Subject Classification 2000
Primary: 05C99, 05E99, 15A06, 54H10, 57N05
Secondary: 15A99, 54E35
##### Milestones
Received: 2 August 2010
Revised: 21 December 2010
Accepted: 22 December 2010
Published: 6 January 2011

Proposed: Chi-Kwong Li
Communicated by Chi-Kwong Li
##### Authors
 Liam Solus Department of Mathematics Oberlin College OCMR 2293 135 W Lorain Street Oberlin, OH 44074 United States