arxiv: 2605.03523 · v1 · submitted 2026-05-05 · 🧮 math.LO

Recognition: unknown

Free sets, thin sets and rainbows for barriers

Lorenzo Carlucci, Oriola Gjetaj

Pith reviewed 2026-05-07 12:45 UTC · model grok-4.3

classification 🧮 math.LO

keywords free setsthin setsrainbow Ramsey theorembarrierscomputabilityreverse mathematicsRamsey theory

0 comments

The pith

Friedman's free-set, thin-set and rainbow theorems extend to colorings of barriers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper formulates and proves generalizations of Friedman's free set theorem, thin set theorem, and the rainbow Ramsey theorem that apply when the underlying colorings are defined on barriers rather than on n-tuples or sequences. It then measures the logical and computational strength of the new statements by establishing upper and lower bounds on the Turing degrees of solutions to computable instances and by exhibiting uniform computable reductions between the problems. These bounds immediately yield corollaries about the proof-theoretic strength of the statements inside subsystems of second-order arithmetic.

Core claim

We formulate and prove the generalizations of Friedman's free set and thin set theorems and of the rainbow Ramsey theorem to colorings of barriers. We analyze the strength of these theorems from the point of view of computability theory proving some upper and lower bounds on the complexity of solutions for computable instances and some uniform computable reductions. We obtain as corollaries some proof-theoretical results on the logical strength of the theorems, in the spirit of reverse mathematics.

What carries the argument

Barriers, the combinatorial objects that serve as the new domain for the colorings and that support the definitions of free sets, thin sets, and rainbow sets in the generalized statements.

If this is right

Every computable barrier coloring admits a free set, thin set, or rainbow set whose Turing degree is bounded by a fixed computable degree.
Uniform computable reductions hold between the generalized free-set, thin-set, and rainbow problems.
The generalized statements are equivalent to specific subsystems of second-order arithmetic, fixing their exact proof-theoretic strength.
Lower-bound constructions show that some computable instances require solutions of positive degree.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same barrier technique may permit uniform generalizations of other classical Ramsey statements whose computability and proof-theoretic strengths are already classified.
Barriers could supply a common setting in which to compare the Weihrauch degrees of these problems with those of related infinitary combinatorial principles.
The results suggest that barrier colorings might be used to calibrate the strength of statements that lie strictly between arithmetic comprehension and full Ramsey theory.

Load-bearing premise

The chosen definition of barriers produces non-trivial generalizations for which the stated existence theorems and computability bounds hold exactly as formulated.

What would settle it

A single computable coloring of a barrier whose every free set, thin set, or rainbow set lies outside the claimed degree bound, or for which no uniform computable reduction to the other problems exists, would refute the bounds.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper extends Friedman's free set, thin set, and rainbow Ramsey theorems to barriers and records some computability bounds plus reverse math corollaries.

read the letter

The main takeaway is that the authors take three known Ramsey-type results and prove versions for colorings of barriers, while also giving computability upper and lower bounds on solutions for computable instances and some uniform reductions. The reverse mathematics corollaries follow directly from that analysis in the usual style of the field. This is new relative to the classical statements on the naturals, since barriers are a distinct combinatorial object and the effective content had not been checked before. The work fills in these cases cleanly without adding extra machinery. It does what it sets out to do by stating the generalizations, proving existence, and extracting the computability facts. The presentation stays within standard techniques for this corner of logic, which keeps the contribution focused and easy to locate in the literature. The soft spots are minor and proportional to the scope. The results follow the expected pattern for such extensions, so they add to the collection of principles rather than revealing unexpected behavior. The bounds are concrete but the paper does not claim they are optimal, which is reasonable. The definition of barriers works for the proofs, though a referee might ask for more discussion of why this particular notion is the right one to generalize. Nothing indicates circularity or unsupported claims. This paper is for people already working in reverse mathematics and computability theory on infinitary combinatorics. A reader who knows Friedman's theorems and rainbow Ramsey will find the barrier cases and the complexity bounds useful to have on record. Someone outside that subfield will not get much out of it. The paper is grounded enough and the claims specific enough to deserve a serious referee. I would recommend sending it for peer review.

Referee Report

0 major / 3 minor

Summary. The paper formulates and proves generalizations of Friedman's free-set and thin-set theorems together with the rainbow Ramsey theorem to colorings of barriers. It establishes computability-theoretic upper and lower bounds on the complexity of solutions to computable instances, identifies uniform computable reductions, and derives corresponding reverse-mathematics corollaries on logical strength.

Significance. If the stated generalizations and bounds hold, the work supplies a coherent extension of three classical Ramsey-type results to the combinatorial setting of barriers, accompanied by explicit computability and proof-theoretic information. The uniform reductions and reverse-mathematics corollaries constitute concrete strengths that locate the new theorems inside the existing hierarchy.

minor comments (3)

[Introduction] The abstract and introduction should include a brief, self-contained definition or reference for the notion of barrier used throughout (e.g., the precise closure properties under initial segments and the Nash-Williams property).
[Computability analysis] In the computability section, the precise Turing-degree bounds (e.g., whether solutions are always computable from 0' or require higher jumps) should be stated uniformly for all three theorems rather than scattered across separate subsections.
[Proof-theoretic results] The reverse-mathematics corollaries would benefit from an explicit comparison table listing the exact subsystems (e.g., ACA0, ATR0, or Π11-CA0) to which each generalized statement is equivalent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the accurate summary of its contributions, and the recommendation for minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point rebuttal or clarification at this time. Any minor editorial or presentational suggestions will be incorporated in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper directly formulates and proves generalizations of Friedman's free-set/thin-set theorems and the rainbow Ramsey theorem to barrier colorings, then derives computability upper/lower bounds and reverse-mathematics corollaries via standard proof techniques. No load-bearing step reduces by construction to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain whose cited result itself depends on the target claim. All existence statements and strength analyses rest on explicit combinatorial arguments and known external results whose verification is independent of the present derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no specific free parameters, axioms, or invented entities can be extracted. The work presumably relies on standard set-theoretic axioms and prior definitions of barriers and the classical theorems.

pith-pipeline@v0.9.0 · 5360 in / 1045 out tokens · 46360 ms · 2026-05-07T12:45:08.315693+00:00 · methodology

Review history (4 revisions) →

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 1 canonical work pages

[1]

4, 89–106 (fr)

Marc Assous,Caractérisation du type d’ordre des barrières de Nash-Williams, Publications du Département de mathématiques (Lyon)11(1974), no. 4, 89–106 (fr). MR 366758

1974
[2]

Lorenzo Carlucci, Oriola Gjetaj, Quentin Le Houérou, and Ludovic Levy Patey,Ramsey-like theorems for the Schreier barrier, J. Symb. Log. (2025)

2025
[4]

Lorenzo Carlucci and Konrad Zdanowski,The strength of Ramsey’s theorem for coloring relatively large sets, J. Symb. Log.79(2014), no. 1, 89–102. MR 3226013 FREE SETS, THIN SETS AND RAINBOWS FOR BARRIERS 13

2014
[5]

Raphaël Carroy and Yann Pequignot,From well to better, the space of ideals, Fundam. Math. 227(2014), no. 3, 247–270

2014
[6]

Peter Cholak and Ludovic Patey,Thin set theorems and cone avoidance, Trans. Amer. Math. Soc.373(2020), no. 4, 2743–2773. MR 4069232

2020
[7]

Cholak, Mariagnese Giusto, Jeffry L

Peter A. Cholak, Mariagnese Giusto, Jeffry L. Hirst, and Carl G. Jockusch, Jr.,Free sets and reverse mathematics, Reverse mathematics 2001, Lect. Notes Log., vol. 21, Assoc. Symbol. Logic, La Jolla, CA, 2005, pp. 104–119. MR 2185429

2001
[8]

Peter Clote,A recursion theoretic analysis of the clopen Ramsey theorem, J. Symb. Log.49 (1984), no. 2, 376–400 (en)

1984
[9]

,A generalization of the limit lemma and clopen games, J. Symb. Log.51(1986), no. 2, 273–291. MR 840405

1986
[10]

Csima and Joseph R

Barbara F. Csima and Joseph R. Mileti,The strength of the rainbow Ramsey theorem, J. Symb. Log.74(2009), no. 4, 1310–1324

2009
[11]

François Dorais, Damir Dzhafarov, Jeffry Hirst, Joseph Mileti, and Paul Shafer,On uniform relationships between combinatorial problems, Trans. Amer. Math. Soc.368(2015), no. 2, 1321–1359 (en)

2015
[12]

Dzhafarov and Carl Mummert,Reverse Mathematics: Problems, Reductions, and Proofs, Theory and Applications of Computability, Springer International Publishing, Cham, 2022 (en)

Damir D. Dzhafarov and Carl Mummert,Reverse Mathematics: Problems, Reductions, and Proofs, Theory and Applications of Computability, Springer International Publishing, Cham, 2022 (en)

2022
[13]

Fred Galvin and Karel Prikry,Borel sets and Ramsey’s theorem, J. Symb. Log.38(1973), 193–198. MR 0337630

1973
[14]

Jockusch,Ramsey’s theorem and recursion theory, J

Carl G. Jockusch,Ramsey’s theorem and recursion theory, J. Symb. Log.37(1972), no. 2, 268–280 (en)

1972
[15]

Lu Liu and Ludovic Patey,The reverse mathematics of the Thin set and Erdős-Moser theorems, J. Symb. Log.87(2022), no. 1, 313–346

2022
[16]

Notes Log., vol

Alberto Marcone,Wqo and bqo theory in subsystems of second order arithmetic, Reverse mathematics 2001, Lect. Notes Log., vol. 21, Assoc. Symbol. Logic, La Jolla, CA, 2005, pp. 303–330. MR 2185443

2001
[17]

Alberto Marcone, Antonio Montalbán, and Andrea Volpi,The barrier Ramsey theorem, (2025), arXiv:2505.02544

work page arXiv 2025
[18]

Crispin St. J. A. Nash-Williams,On better-quasi-ordering transfinite sequences, Proc. Cam- bridge Philos. Soc.64(1968), 273–290. MR 221949

1968
[19]

Math.216(2016), no

Ludovic Patey,The weakness of being cohesive, thin or free in reverse mathematics, Israel J. Math.216(2016), no. 2, 905–955. MR 3557471

2016
[20]

Maurice Pouzet,Sur les prémeilleurordres, Ann. Inst. Four.22(1972), no. 2, 1–19

1972
[21]

Hartley Rogers,Theory of recursive functions and effective computability, 1st mit press ed., MIT Press, Cambridge, Mass, 1987

1987
[22]

Simpson,Subsystems of second order arithmetic, second ed., Perspectives in Logic, Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie, NY,

Stephen G. Simpson,Subsystems of second order arithmetic, second ed., Perspectives in Logic, Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie, NY,
[23]

Clifford Spector,Recursive well-orderings, J. Symb. Log.20(1955), no. 2, 151–163 (en)

1955
[24]

Stevo Todorcevic,Introduction to ramsey spaces, Princeton University Press, Princeton, 2010

2010
[25]

Math.261(2014), 1–25

Wei Wang,Some logically weak Ramseyan theorems, Adv. Math.261(2014), 1–25. MR 3213294

2014