Gabriela Laboska : Some Computabiity-theoretic Aspects of Partition Regularity over Algebraic Structures

Posted by Matthew Harrison-Trainor , part of the Logic Seminar.

At

Oct. 1, 2024, 4 p.m.

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Abstract

An inhomogeneous system of linear equations over a ring $R$ is partition regular if for any finite coloring of $R$, the system has a monochromatic solution. In 1933, Rado showed that an inhomogeneous system is partition regular over $\mathbb{Z}$ if and only if it has a constant solution. Following a similar approach, Byszewski and Krawczyk showed that the result holds over any integral domain. In 2020, Leader and Russell generalized this over any commutative ring $R$, with a more direct proof than what was previously used. We analyze some of these combinatorial results from a computability-theoretic point of view, starting with a theorem by Straus used directly or as a motivation to many of the previous results on the subject.