Multiply partition regular matrices
Dennis Davenport, Neil Hindman, Imre Leader, and Dona Strauss

TL;DR
This paper characterizes when pairs of matrices are doubly kernel partition regular, showing a necessary and sufficient condition involving the existence of a positive rational scalar making a combined matrix kernel partition regular.
Contribution
It establishes a precise criterion for doubly kernel partition regularity of matrix pairs, extending to multiple matrices, thus advancing the understanding of partition regularity in combinatorics.
Findings
A pair of matrices is doubly kernel partition regular if and only if a scaled combined matrix is kernel partition regular.
The criterion extends to multiple matrices, providing a comprehensive framework.
The results unify and generalize previous partial conditions for partition regularity.
Abstract
Let be a finite matrix with rational entries. We say that is {\it doubly image partition regular\/} if whenever the set of positive integers is finitely coloured, there exists such that the entries of are all the same colour (or {\it monochromatic\/}) and also, the entries of are monochromatic. Which matrices are doubly image partition regular? More generally, we say that a pair of matrices , where and have the same number of rows, is {\it doubly kernel partition regular\/} if whenever is finitely coloured, there exist vectors and , each monochromatic, such that . There is an obvious sufficient condition for the pair to be doubly kernel partition regular, namely that there exists a positive rational such that the matrix…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
Topicsgraph theory and CDMA systems · Rings, Modules, and Algebras · Advanced Topics in Algebra
