Increasing Polarized 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. A colouring $f:[\mathbb N]^2\to\{0,1\}$ assigns a colour to each unordered pair; write $f(x,y)$ for $x<y$. The principle $\mathsf{RT}^2_2$ asserts an infinite homogeneous set $H$. A colouring is stable if for every $x$ the values $f(x,y)$ are constant for all sufficiently large $y$; $\mathsf{SRT}^2_2$ restricts $\mathsf{RT}^2_2$ to stable colourings. The increasing polarized principle $\mathsf{IPT}^2_2$ asserts infinite $H_1,H_2\subseteq\mathbb N$ such that all $f(x,y)$ with $x\in H_1$, $y\in H_2$ and $x<y$ have one colour.

Statement recorded in the supplied source.

Is $\mathsf{IPT}^2_2$ equivalent to $\mathsf{RT}^2_2$, to $\mathsf{SRT}^2_2$, to both, or to neither? [S2]

Short English statement

Does the increasing polarized Ramsey principle have the same strength as the ordinary two-colour pair theorem, its stable version, both, or neither?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7500108 (AMR-074-0108). 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 8, printed p. 4. https://math.berkeley.edu/~antonio/papers/questionsRM.pdf.