Contributed Talk

# Some computability-theoretic aspects of partition regularity over rings

## Gabriela Laboska

Friday, 16:55, J336

A 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 [3] 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 [1] showed that the result holds over any integral domain. In 2018, following a different approach, Leader and Russell [2] generalized this over any commutative ring \(R\). We analyze some of these combinatorial results from a computability-theoretic point of view, starting with a theorem by Straus [4] used in the work of [1] and [2] that generalizes an earlier result by Rado.

## Bibliography

- Byszewski, B., Krawczyk, E.,
*Rado’s theorem for rings and modules*,,vol. 180 (2021), no. 105402, pp. 28.*Journal of Combinatorial Theory Series A* - Leader, I., Russell, P. A.,
*Inhomogeneous partition regularity*,,vol. 27 (2020), no. 2, pp. 4.*Electronic Journal of Combinatorics* - Rado, R.,
*Studien zur kombinatorik*,,vol. 36 (1933), no. 1, pp. 424–470.*Mathematische Zeitschrift* - Straus, E. G.,
*A combinatorial theorem in group theory*,,vol. 29 (1975), pp. 303-309.*Mathematics of Computation*