Strength of the Dual Ramsey Theorem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Work in second-order arithmetic over $\mathsf{RCA}_0$, the base theory with $\Delta^0_1$ comprehension and $\Sigma^0_1$ induction. The strength of a principle means which subsystems prove it and which subsystems it implies over that base; equivalence requires both implications. For a positive finite $k$ or $k=\omega$, $(\omega)^k$ is the space of partitions of $\mathbb N$ into $k$ nonempty pieces, with its usual product coding by assigning each integer its block label, labels ordered by the least member of each block. A Borel colouring is a finite-valued colouring whose colour classes are Borel in this coding. If $X\in(\omega)^\omega$, let $(X)^k$ be the partitions into $k$ pieces coarser than $X$, so every piece is a union of pieces of $X$. The dual Ramsey principle $\mathsf{DRT}_k$ says that every finite Borel colouring of $(\omega)^k$ has an $X\in(\omega)^\omega$ for which $(X)^k$ is monochromatic.

Statement recorded in the supplied source.

Determine the reverse-mathematical strength of $\mathsf{DRT}_k$, for $k\leq\omega$. [S2]

Short English statement

Determine which set-existence axioms are required for the Borel dual Ramsey theorem for partitions into finitely many, or countably many, pieces.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7500110 (AMR-074-0110). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Antonio Montalbán, Open Questions in Reverse Mathematics, author preprint dated 17 August 2010, Question 10, printed p. 5. https://math.berkeley.edu/~antonio/papers/questionsRM.pdf.