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
, the metric
space (,
). Here,
is the set of all matrices
equivalent to
that are
(not) partition regular if
is (not) partition regular; and for elementary matrices,
and
, we
let
.
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.
