Vol. 3, No. 4, 2010

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A topological generalization of partition regularity

Liam Solus

Vol. 3 (2010), No. 4, 421–433

In 1939, Richard Rado showed that any complex matrix is partition regular over 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 Φ, the metric space (MΦ, δ). Here, MΦ is the set of all matrices equivalent to Φ that are (not) partition regular if Φ is (not) partition regular; and for elementary matrices, Ei and Fj, we let δ(A,B) = min{m = l + k : B = E1ElAF1Fk}. 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.

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
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
Liam Solus
Department of Mathematics
Oberlin College
OCMR 2293
135 W Lorain Street
Oberlin, OH 44074
United States