Image partition regularity over the integers, rationals and reals

New York Journal of Mathematics
New York J. Math. 11 (2005) 519–538.
Image partition regularity over the integers,
rationals and reals
Neil Hindman and Dona Strauss
Abstract. There is only one reasonable definition of kernel partition regular-
ity over any subsemigroup of the reals. On the other hand, there are several
reasonable definitions of image partition regularity. We establish the exact
relationships that can hold among these various notions for finite matrices and
infinite matrices with rational entries. We also introduce some hybrid notions
and describe their relationship to what is probably the major unsolved prob-
lem in kernel partition regularity, namely whether an infinite matrix which is
kernel partition regular over Q must be kernel partition regular over N.
Contents
1. Introduction 519
+ +2. Image partition regularity over N, Z, Q , Q, R and R 523
3. Connections between image and kernel partition regularity 529
References 538
1. Introduction
Image partition regularity is one of the most important concepts of Ramsey
Theory. Suppose that A is a finite or infinite matrix over Q in which there are
only a finite number of nonzero entries in each row. A is said to be image partition
regular over the set N of positive integers, if given any finite partition of N, there
is a vector x, with entries in N, such that Ax is defined and all the entries of Ax
lie in the same cell of the partition.
The significance of this concept can be illustrated by considering some of ...