Ramsey's Theorem for Triples over a Weak Base

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.

Let $[\mathbb N]^3$ be the three-element subsets of $\mathbb N$. Write $\mathsf{RT}^3_2$ for: every colouring $c:[\mathbb N]^3\to\{0,1\}$ has an infinite $H\subseteq\mathbb N$ on which $c$ is constant. $\mathsf{RCA}_0^*$ has $\Delta^0_1$ comprehension, bounded ($\Sigma^0_0$) induction and total exponentiation; adding $\Sigma^0_1$ induction yields $\mathsf{RCA}_0$. $\mathsf{ACA}_0$ adds arithmetical comprehension to $\mathsf{RCA}_0$.

Statement recorded in the supplied source.

Does the familiar equivalence of Ramsey's theorem for triples with $\mathsf{ACA}_0$ continue to hold over $\mathsf{RCA}_0^*$, that is, does this weaker base prove $\mathsf{RT}^3_2\leftrightarrow\mathsf{ACA}_0$? This records the historical comparison; the later source treats its weak-induction setting in detail. [S3] [S2]

Short English statement

Does reducing induction in the base theory preserve the usual equivalence between Ramsey's theorem for triples and arithmetical comprehension?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7500216 (AMR-074-0216). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Stephen G. Simpson, Open Problems in Reverse Mathematics (15 December 1999), Replacing RCA0 by a Weaker Base Theory, paragraph beginning One project for the future. https://sgslogic.net/t20/talks/cta/problems/node9.html.
• S3 Leszek Aleksander Kołodziejczyk, Katarzyna Kowalik and Keita Yokoyama, How Strong Is Ramsey's Theorem If Infinity Can Be Weak?, arXiv:2011.02550v2 (16 January 2021), Introduction and Section 3. https://arxiv.org/pdf/2011.02550.