Open Collections

Description

We will consider two families of partition principles parameterized by dimension: Galvin's problem in dimension $d$ ($\mathsf{G}_d$) as studied by Raghavan and Todorcevic (2023), and the Polish grid principle in dimension $d$ ($\mathsf{PG}_d$) as studied by Lambie-Hanson and Zucker (2024). Both $\mathsf{G}_d$ and $\mathsf{PG}_d$ fail in models of $\mathsf{ZFC}+2^{\aleph_0}\leq\aleph_{d-2}$. We introduce a family of partition principles for colouring spaces of unordered products that generalizes both $\mathsf{G}_d$ and $\mathsf{PG}_d$, and recovers the common bound.