A new mathematical preprint reports a formal result in reverse mathematics: for every natural number n and every ℓ at least 1, the thin-set principle RT^n_{<∞,ℓ} implies the Σ^0_{n+1}-bounding principle BΣ^0_{n+1} over RCA_0. In plain language, a coloring statement with a restricted color range formally yields a bounding statement in the same base theory.
For n and ℓ at least 1, RT^n_{<∞,ℓ} says that every coloring of the n-element subsets of the natural numbers has an infinite set H on which the coloring uses at most ℓ colors. The phrase thin set refers to that restricted palette, not to a physical sample or population.
A theorem about formal strength
The question sits within reverse mathematics of second-order arithmetic, a field that compares formal statements by asking what they can derive from a fixed base. Here, that base is RCA_0, and the study asks whether the thin-set theorem yields BΣ^0_{n+1} at every permitted n and ℓ. There are no participants or observational data; the objects of analysis are principles, colorings, functions, and infinite sets.
The central theorem has an intermediate step. For any n and ℓ at least 1, RT^n_{<∞,ℓ} first yields the corresponding one-dimensional Σ^0_n thin-set principle over RCA_0. The argument then derives BΣ^0_{n+1} from the original thin-set principle, linking the two levels of the formal system.
Inside the proof
The proof establishes both main implications simultaneously by induction on the standard natural number n, with n at least 1. It invokes a limit lemma over RCA_0 + BΣ^0_n for total Σ^0_n functions with finite range. A separate approximation-transfer lemma concludes that an ℓ-homogeneous set for an approximating function g is also ℓ-homogeneous for the function φ.
A further theorem places the result in a wider equivalence. Over RCA_0, BΣ^0_{n+1} is equivalent to both the unrestricted weakened one-dimensional thin-set principle and its ℓ-bounded version, for n and ℓ at least 1. Within this formal setting, the bounding principle and those weakened thin-set statements therefore have the same deductive strength.
A sharper picture for pairs
The paper draws a specific consequence for pairs, meaning two-element subsets. For every ℓ at least 1, the first-order part of RCA_0 + RT^2_{<∞,ℓ} is the same as that of RCA_0 + BΣ^0_3. This compares the number-only consequences of the two formal systems.
Another corollary marks a boundary between two coloring principles. For every ℓ at least 1, RCA_0 does not prove that the two-color pair theorem RT^2_2 implies RT^2_{<∞,ℓ}. The statement concerns provability inside the base theory, not a particular coloring that was tested or a numerical failure that was observed.
What remains unresolved
The result establishes a lower bound on the formal strength captured by the thin-set theorem, but it does not settle the optimal induction level. The paper leaves open whether, for n and ℓ at least 1 with d_{n−1} smaller than ℓ, RCA_0 does not prove the implication RT^n_{<∞,ℓ} → IΣ^0_{n+1}.
The authors also leave open a color-reduction question. For n and ℓ at least 1, is there a natural number m such that RCA_0 proves RT^{n+1}_{ℓ+1,ℓ} → RT^n_{<∞,m}? The question asks whether a principle at one dimension and color setting can yield a thin-set principle one dimension lower with some finite color bound.
A formal result, not an empirical study
The manuscript is an arXiv version 1 preprint dated 26 Aug 2026. Its evidence consists of formal derivability claims and proof arguments, and no empirical sample is reported. The result concerns the formal strength of combinatorial principles rather than a human, clinical, or applied outcome.
Paper data and sources
Original title: Thin set theorem for arbitrarily many colors implies bounding
Authors: Yamato Miyata, Keita Yokoyama
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text