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Rainbow Ramsey theorem for triples is strictly weaker than the Arithmetic Comprehension Axiom

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Wei Wang

Logic · Submitted 14 Mar 2013

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that $\RCA + \RRT^3_2 \not\vdash \ACA$ where $\RRT^3_2$ is the Rainbow Ramsey Theorem for 2-bounded colorings of triples.

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arXiv ID1303.3327 (version 1)
Pith IDVMQB5HNQ, Pith's permanent record of this paper and its review
CitationWei Wang. "Rainbow Ramsey theorem for triples is strictly weaker than the Arithmetic Comprehension Axiom." arXiv:1303.3327 (2013). https://pith.science/paper/VMQB5HNQ
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,038 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.74 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.20 · compares with prior results 0.47 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We prove that $\RCA + \RRT^3_2 \not\vdash \ACA$ where $\RRT^3_2$ is the Rainbow Ramsey Theorem for 2-bounded colorings of triples. This reverse mathematical result is based on a cone avoidance theorem, that every 2-bounded coloring of pairs admits a cone-avoiding infinite rainbow, regardless of the complexity of the given coloring. We also apply the proof of the cone avoidance theorem to the question whether $\RCA + \RRT^4_2 \vdash \ACA$ and obtain some partial answer.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove that $\RCA + \RRT^3_2 \not\vdash \ACA$ where $\RRT^3_2$ is the Rainbow Ramsey Theorem for 2-bounded colorings of triples. result 0.98 0.95 0.95 asserted 1.00
s1 This reverse mathematical result is based on a cone avoidance theorem, that every 2-bounded coloring of pairs admits a cone-avoiding infinite rainbow, regardless of the complexity of the given coloring. method 0.85 0.74 0.08 asserted 1.00
s2 We also apply the proof of the cone avoidance theorem to the question whether $\RCA + \RRT^4_2 \vdash \ACA$ and obtain some partial answer. contribution 0.50 0.94 0.04 asserted 0.78

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We prove that $\RCA + \RRT^3_2 \not\vdash \ACA$ where $\RRT^3_2$ is the Rainbow Ramsey Theorem for 2-bounded colorings of triples. This reverse mathematical result is based on a cone avoidance theorem, that every 2-bounded coloring of pairs admits a cone-avoiding infinite rainbow, regardless of the complexity of the given coloring. We also apply the proof of the cone avoidance theorem to the question whether $\RCA + \RRT^4_2 \vdash \ACA$ and obtain some partial answer.

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