# Logic Seminar: Some Computabiity-theoretic Aspects of Partition Regularity over Algebraic Structures, by Gabriela Laboska

October 1, 2024

4:00 PM - 4:50 PM

Gabriela Laboska (University of Chicago): Some Computabiity-theoretic Aspects of Partition Regularity over Algebraic Structures

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.

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## Date posted

Oct 7, 2024

## Date updated

Oct 7, 2024