Rado's Theorem on vector space

Document Type : Original Article

Authors
Ghadir Ghadimi 1
Mohammad Akbari Tootkaboni 2
1 Department of Pure Mathematics,Faculty of Mathematical Sciences, guilan, Iran.
2 Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
10.22034/maco.5.2.8
Abstract
In this paper, we investigate partition regularity of matrices over vector spaces where the matrix entries consist of linear transformations on the vector space. The paper also examines the existence of monochromatic solutions for matrix entries of linear transformations, which generalizes Rado's theorem.Our main results establish that matrices satisfying these conditions are not only partition regular but also preserve solutions within large combinatorial sets, such as central sets. Furthermore, we present a counterexample involving affine maps, demonstrating a limitation of these results and highlighting the necessity of the linearity assumption. This work bridges combinatorial number theory and linear algebra, offering a functional-analytic perspective on Ramsey-type problems.

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Mathematical Analysis and Convex Optimization
Volume 5, Issue 2
December 2025