Image partition regularity over the reals

New York Journal of Mathematics
New York J. Math. 9 (2003) 79–91.
Image partition regularity over the reals
Neil Hindman
Abstract. We show that many of the natural analogues of known character-
izations of image partition regularity and weak image partition regularity of
matrices with rational entries over the integers are valid for matrices with real
entries over the reals.
Contents
1. Introduction 79
2. Preliminary results 82
3. Weak image partiton regularity over R 86
+4. Image partition regularity over R 88
References 91
1. Introduction
In 1933 R. Rado published [8] his famous theorem characterizing those finite
matrices A with rational entries that have the property that whenever N is finitely
colored, there must be some x in the kernel of A all of whose entries are the same
color (or monochrome). This characterization was in terms of the columns condition
which we shall describe below.
In 1943 Rado published a paper [9], among whose results was the fact that the
same condition characterized those finite matrices with real entries that have the
property that whenever R is finitely colored, there is some x in the kernel of A
whose entries are monochrome.
+
Definition 1.1. Let u, v∈ N, let S∈{N,Z,R ,R}. Let F = Q if S = N or S = Z,
+and let F = R if S = R = {x ∈ R : x>0} or S = R. Let A be a u× v matrix
with entries from F.
Received August 19, 2002.
Mathematics Subject Classification. 05D10.
Key words and phrases. Ramsey Theory, partition regular, matrices, Rado’s Theorem, ...