Bounded Ramsey's theorem for triples in computability theory

Abstract: We study a restriction of Ramsey's theorem for 2-coloring of triples, in which homogeneous sets for color~1 are of bounded size ($\mathsf{BRT}^3_2$). We prove that the computational content of this statement is very close to Ramsey's theorem for pairs ($\mathsf{RT}^2_2)$, in that it satisfies the same known computability-theoretic upper bounds, but that $\mathsf{BRT}^3_2$ is not computably-reducible to $\mathsf{RT}^2_2$, even when allowing multiple applications of $\mathsf{RT}^2_2$.

• The paper demonstrates that BRT³₂ exhibits key computability features such as cone avoidance and a weakly low basis, aligning with RT²₂ on several fronts.
• It applies game-theoretic reducibility and hyperimmunity preservation techniques to show sharp separations, notably that BRT³₂ is not computably reducible to RT²₂.
• The work bridges reverse mathematics and computability theory by using cohesion and combinatorial forcing to derive effective arithmetical bounds and structural decompositions.

Bounded Ramsey's Theorem for Triples: Computability-Theoretic Analysis and Separations

Introduction and Motivation

The paper "Bounded Ramsey's theorem for triples in computability theory" (2604.02092) presents a comprehensive computability-theoretic analysis of a bounded variant of Ramsey's theorem for triples—denoted $BRT^3_2$—within the frameworks of reverse mathematics and computable reducibility. The focus is on colorings $f : [\mathbb{N}]^3 \to 2$, for which homogeneous sets for color 1 are bounded in size, specifically, $BRT^n_{2,\ell}$ asserts that if $f$ has no homogeneous set of size $\ell$ for color 1, then there is an infinite homogeneous set for color 0.

The principal aim is to locate the computability-theoretic strength of $BRT^3_2$ relative to established Ramsey-type principles such as $RT^2_2$ (Ramsey's theorem for pairs with two colors). The analysis sharply addresses upper bounds, basis theorems, cone and trace avoidance, as well as parameterized separation results. Special attention is given to notions of computable reducibility via game semantics, which reflect the complexity of reductions between $\Pi^1_2$ principles.

Structural Framework and Analytical Tools

The study leverages several foundational concepts:

• Bounded Ramsey's theorem ($BRT$): The focus is on the situation where infinite homogeneous sets are guaranteed for one color if the other color's homogeneous sets are uniformly bounded.
• Reverse mathematics: Results are calibrated over $\mathsf{RCA}_0$, with attention to $f : [\mathbb{N}]^3 \to 2$0-models and correspondences to Turing ideals.
• Computable reducibility: The Hirschfeldt-Jockusch reduction game framework is central; $f : [\mathbb{N}]^3 \to 2$1 denotes that $f : [\mathbb{N}]^3 \to 2$2 is reducible to $f : [\mathbb{N}]^3 \to 2$3 over all $f : [\mathbb{N}]^3 \to 2$4-models, with a finitary variant, $f : [\mathbb{N}]^3 \to 2$5, quantifying the number of allowed applications.
• Cohesiveness and stability: Decompositions akin to those of $f : [\mathbb{N}]^3 \to 2$6 are adapted, showing that analysis of $f : [\mathbb{N}]^3 \to 2$7 computability-theoretically reduces to questions about $f : [\mathbb{N}]^3 \to 2$8 instances of $f : [\mathbb{N}]^3 \to 2$9.
• Weakness properties and preservation: Notions such as cone avoidance, constant-bound trace avoidance, and (weakly/strongly) low basis theorems are encoded as preservation properties.
• Diagonalization using hyperimmunities: The fine structure of preservation of $BRT^n_{2,\ell}$0-many hyperimmunities is harnessed to establish parameterized impossibility results for computable reductions.

Upper Bounds: Basis Theorems, Avoidance, and Effectiveness

Several major theorems demonstrate that $BRT^n_{2,\ell}$1 matches $BRT^n_{2,\ell}$2 in terms of favorable computable-theoretic properties:

• Cone avoidance: $BRT^n_{2,\ell}$3 admits cone avoidance, inheriting this property from the Erdős-Moser theorem and cohesiveness via a reduction framework. This confirms that $BRT^n_{2,\ell}$4 is strictly weaker than arithmetic comprehension, as every solution can avoid computing the halting set.
• Constant-bound trace avoidance: $BRT^n_{2,\ell}$5 admits c.b.-trace avoidance; no instance computably traces every path through a given $BRT^n_{2,\ell}$6 class with bounded width. This yields that $BRT^n_{2,\ell}$7 cannot imply $BRT^n_{2,\ell}$8.
• Weakly low basis theorem: Every computable instance of $BRT^n_{2,\ell}$9 admits a solution whose jump is bounded by a chosen PA degree over $f$0. This places $f$1 in line with $f$2, which is known not to admit a genuinely low basis but does have a weakly low basis (the jump is computable in any PA degree over $f$3).
• Effective bounds in the arithmetic hierarchy: Every computable instance of $f$4 has a $f$5 solution, with the impossibility (sharpness) result that some computable instance lacks a $f$6 solution.

These upper bounds are established using nontrivial functional interpretations of the Erdős-Moser theorem, strong preservation theorems, and intricate combinatorial forcing constructions.

Structural Decomposition via Cohesiveness and Erdős-Moser

The manuscript provides an explicit connection between higher-arity stable bounded Ramsey instances and lower-arity instances via Shoenfield's Limit Lemma and the cohesiveness principle. Any computable coloring of triples can be restricted to a stable coloring on a cohesive set, then analyzed as a $f$7 instance of $f$8.

The reduction pipeline is as follows:

1. Apply cohesiveness to achieve stability on a set.
2. Consider the induced limit coloring ($f$9) as a $\ell$0 instance.
3. Solve the instance with $\ell$1-type arguments, yielding an infinite homogeneous set for color 0.

This approach encapsulates the computational content of $\ell$2 and allows direct transfer of avoidance and lowness results from lower-dimensional principles.

Separations: Impossibility Results and Non-Reducibility

Despite this alignment in upper bounds, the paper delivers sharp separations for computable reducibility:

• Iterated reductions are insufficient: For any fixed $\ell$3, $\ell$4. That is, no reduction using a bounded number of applications of $\ell$5 (of any arity) can solve all instances of $\ell$6. The proof utilizes triangle-free graphs of high chromatic number (Erdős-type constructions) to construct colorings maximizing the diagonalization power (through preservation of hyperimmunity failure).
• No computable reduction: The stronger result $\ell$7 is established. There exists a computable instance of $\ell$8 (effectively, a triangle-free $\ell$9 graph) such that no computable reduction can uniformly map any instance of $BRT^3_2$0 (regardless of the number of colors) to a solution set that computes a solution to the $BRT^3_2$1 instance.

The separation arguments are refined, exploiting the structure of triangle-free graphs and the inability of $BRT^3_2$2 (even iterated) to simultaneously preserve sufficiently many hyperimmunities.

Implications and Theoretical Consequences

The analysis provided has several ramifications:

• Refined calibration of combinatorial principles: The results corroborate that tight computability-theoretic hierarchies exist among Ramsey-type principles, and that the effect of bounding homogeneous sets is highly nontrivial—lowering the complexity of the principle but not collapsing it to the lower-dimensional counterpart.
• Formal intransitivity of reductions: The negative results for iterated and computable reducibility underscore the limits of certain forms of proof mining and automated reductions in the context of reverse mathematics, relevant to program extraction and proof complexity.
• Implications for model theory and logic: The preservation results on cones, c.b.-traces, lowness, and hyperimmunity provide templates for analyzing the strength and weakness of other combinatorial and combinatorially-inspired statements, particularly in relation to $BRT^3_2$3, $BRT^3_2$4, and coding of arithmetical properties.
• Potential for automatic reasoning parameterizations: From a broader perspective in logic, the game-theoretic analysis of reductions (Hirschfeldt-Jockusch style games) yields new approaches for computational proof search and classifies which principles can and cannot be used as computational or proof-theoretic oracles.

Conclusion

This work situates the bounded Ramsey's theorem for triples precisely in the computability-theoretic landscape, demonstrating that while $BRT^3_2$5 matches $BRT^3_2$6 with respect to many positive computational properties—cone avoidance, trace avoidance, weakly low basis, and bounds in the arithmetical hierarchy—it is strictly stronger in the sense that it is not computably reducible to $BRT^3_2$7, nor can it be solved with any bounded number of applications even when arbitrary arity or coloring is permitted. These results provide an authoritative understanding of the structural complexity of bounded Ramsey-type principles and clarify the nuanced gradations possible in reverse mathematics.

(2604.02092)