Pith. sign in

REVIEW 4 minor 55 references

$\Pi^0_4$ conservation of a Carlson-Simpson lemma for 1-variable words

T0 review · 0 major / 4 minor · reviewed 2026-07-31 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Adding the two-color Carlson-Simpson lemma for 1-variable words does not prove Σ°₂-induction over weak base systems.

desk verdict Solid ∀Π⁰₄-conservation of CSL¹₂ over BΣ₂ that cleanly answers Chong–Li–Wang–Yang on TT²₂ and Henson indivisibility. read the letter →

arxiv 2607.28116 v1 pith:5XNL4NQZ submitted 2026-07-30 math.LO

Quentin Le Houérou , Ludovic Patey This is my paper · ORCID
classification math.LO MSC 03B3003F3505D10
keywords Carlson-SimpsonlemmareversemathematicsΠ⁰₄-conservationparameterizedlargenessHensongraphindivisibilitytreetheoremforpairsBΣ⁰₂variablewords
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that a classical combinatorial statement about colorings of 1-variable words—the Carlson-Simpson lemma for two colors—adds almost no first-order strength when adjoined to ordinary computable mathematics. Precisely, WKL₀ plus that lemma is a ∀Π⁰₄-conservative extension of RCA₀ plus the bounding principle BΣ⁰₂. In plain terms, any ∀Π⁰₄ sentence proved with the lemma was already provable from the weaker system. Two concrete corollaries follow at once: the 2-color indivisibility of the universal triangle-free Henson graph, and the tree theorem for pairs in two colors, both fail to imply Σ⁰₂-induction. That settles a question raised by earlier work on tree partition principles. The argument proceeds by refining the indicator method with a parameterized notion of ordinal largeness that is closed under the relevant finitary Ramsey theorems.

What carries the argument

Parameterized α-largeness (θ) for ordinals of the form ω^n·k, together with finitary closure theorems showing that sufficiently large sets remain large after applications of Graham-Rothschild and a block-homogeneous form of Carlson-Simpson; these feed an indicator construction that produces a semi-regular cut modeling the desired principles while preserving a given ∀Π⁰₄ sentence.

What would settle it

Exhibit a ∀Π⁰₄ sentence that is provable in WKL₀ + CSL¹₂ yet fails in some model of RCA₀ + BΣ⁰₂, or find a gap in the inductive bookkeeping that produces the primitive-recursive largeness bounds for GR¹ and BCSL¹.

Watch

Extended reading notes

Core claim

WKL₀ + CSL¹₂ is a ∀Π⁰₄-conservative extension of RCA₀ + BΣ⁰₂. Consequently neither 2-color indivisibility of the Henson graph H₃ nor the tree theorem TT²₂ implies IΣ⁰₂ over RCA₀.

Load-bearing premise

The finitary combinatorial bounds must hold inside RCA₀: every set that is large enough in the parameterized sense stays large after one step of Graham-Rothschild or block-homogeneous Carlson-Simpson.

Editorial extensions

If this is right

  • WKL₀ + CSL¹₂ does not imply ACA₀ or even IΣ⁰₂.
  • 2-color indivisibility of the universal triangle-free Henson graph is ∀Π⁰₄-conservative over RCA₀ + BΣ⁰₂ and does not prove Σ⁰₂-induction.
  • The tree theorem for pairs and two colors TT²₂ is likewise ∀Π⁰₄-conservative over RCA₀ + BΣ⁰₂, answering Chong-Li-Wang-Yang.
  • The full statement ∀ℓ CSL¹ℓ is not even Π¹-conservative over the same base, since it already yields BΣ⁰₃.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same largeness-plus-indicator template should apply to other unordered variable-word principles whose only known strength comes from an embedded Ramsey theorem for pairs.
  • If the block-homogeneous intermediate principle BCSL¹ can be shown Π¹₁-conservative on its own, the gap between level Carlson-Simpson and full Carlson-Simpson would be isolated entirely in the Ramsey-for-pairs component.
  • The conservation fails as soon as one demands ordered ω-variable words or unboundedly many colors, suggesting a sharp dividing line between unordered two-color and ordered or multi-color variants.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper proves that WKL_{0} + CSL^{1}_{2} is ∀Π^{0}_{4}-conservative over RCA_{0} + BΣ^{0}_{2} (Main Theorem 1.1). The argument proceeds by introducing a parameterized α-largeness notion (θ-apartness), establishing finitary closure of this largeness under the Graham–Rothschild theorem for 1-variable words (Theorem 4.1 via Proposition 4.2) and under a block-homogeneous variant BCSL^{1} of the level Carlson–Simpson lemma (Theorem 5.6 via Proposition 5.7), then feeding those closures into a standard Kirby–Paris-style cut construction that produces a model of WKL_{0} + RT^{2}_{2} + LCSL^{1} (Theorem 6.5). Consequences include that neither 2-color indivisibility of the Henson graph H_{3} nor TT^{2}_{2} implies IΣ^{0}_{2}, answering a question of Chong–Li–Wang–Yang.

Significance. The result cleanly separates CSL^{1}_{2} (and its structural-Ramsey and tree-theoretic consequences) from Σ^{0}_{2}-induction, while remaining consistent with the known lower bound that the fully quantified statement @ℓ CSL^{1}_ℓ already yields BΣ^{0}_{3}. The technical contribution is a workable intermediate invariant—BCSL^{1} block-homogeneity—that keeps the number of homogeneity colors at ℓ^{2n} rather than ℓ^{|X|}, allowing the indicator method to go through for a principle previously known only to sit below ACA_{0}/ACA^{+}_{0}. The applications to Henson-graph indivisibility and TT^{2}_{2} are immediate and settle an explicit open question. The combinatorial core is fully explicit (primitive-recursive towers) and self-contained once earlier GR^{0}/OVW^{0} largeness bounds are taken as black boxes.

minor comments (4)
  1. [§4–§5] The mutual inductive definitions of the towers (b_n, c_n) in the proofs of Theorems 4.1 and 5.6 are dense; a short schematic diagram or a one-line summary of the exponent growth would help the reader track the sparsity and color-count hypotheses.
  2. [Definition 5.4] In Definition 5.4 the clause for ω^{n+1}-block-homogeneity mixes the color of the singleton min X with the color sequence of the remaining set; a brief remark that the resulting color length is still 2n+1 would make the later pigeon-hole applications clearer.
  3. [References] Several citations appear with future dates (e.g., ‘June 2026’, ‘2026’); these should be updated or marked as preprints for the published version.
  4. Typographical inconsistencies such as ‘BΣ0 2’ versus ‘BΣ^{0}_{2}’ and occasional missing spaces around math operators appear throughout; a uniform pass would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: conservation follows from new finitary largeness closures plus a standard indicator cut; self-citations are independent black-box lemmas.

full rationale

The load-bearing chain is: (i) new RCA₀ proofs that sufficiently ω-large(θ) sets are GR¹-ωⁿ-large(θ) (Thm 4.1 via Prop 4.2) and BCSL¹-ωⁿ-large(θ) (Thm 5.6 via Prop 5.7); (ii) the standard Kirby–Paris-style cut construction of §6 that turns those closures plus RT²₂-largeness into a semi-regular cut model of WKL₀+RT²₂+LCSL¹, hence of WKL₀+CSL¹₂ by Lemma 5.3; (iii) transfer of the ∀Π⁰₄ sentence. None of these steps defines the target conservation in terms of itself, fits a parameter to data and renames it a prediction, or imports a uniqueness theorem that already encodes the conclusion. Prior self-citations ([31] for f_GR0/f_OVW0/CSL⁰ bounds; [39,40] for RT²₂-largeness) supply only base combinatorial black boxes whose statements do not presuppose ∀Π⁰₄-conservation of CSL¹₂. The multi-step color-invariance bookkeeping (sparsity x↦x^{x^x}, color-count bounds ℓ^{(2n+1)k²}<min X₀, mutual induction of b_n,c_n, block-homogeneity keeping ≤ℓ^{2n} colors) is an ordinary inductive argument inside RCA₀, not a definitional identity. Score 1 only for routine author-overlap citations that are not circular in the sense of the checklist.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The paper works entirely inside subsystems of second-order arithmetic. All background principles (RCA₀, BΣ₂, WKL₀, finitary GR/OVW bounds, Ketonen-Solovay-style largeness) are standard or previously published; the only new devices are technical definitions (BCSL block-homogeneity, X-variable words) introduced to make the induction go through.

assumptions (4)
  • standard math RCA₀ (Δ⁰₁-comprehension + IΣ⁰₁) as base theory
    Universal base system for reverse mathematics; used throughout.
  • standard math BΣ⁰₂ (bounding for Σ⁰₂ formulas)
    The theory over which conservation is proved; Proposition 3.3 relies on it for largeness.
  • standard math Existence of primitive-recursive bounds for finitary Graham-Rothschild and OVW (Theorems 2.3-2.4)
    Cited from Shelah, Dodos-Kanellopoulos et al.; used to obtain f_GR0, f_OVW0.
  • domain assumption Parameterized α-largeness is a largeness notion under BΣ₂ (Prop. 3.3)
    Taken from the authors' earlier work [39]; load-bearing for the indicator construction.
invented entities (2)
  • BCSL¹-α-block-homogeneity
    purpose: Intermediate finitary invariant between full CSL and level-CSL that keeps the number of monochromatic colors polynomial in n rather than exponential in |X|.
    Defined in Def. 5.4-5.5 solely to close the induction in §5; no external meaning claimed.
  • X-variable word / set(w) largeness
    purpose: Bridge between set-theoretic largeness and variable-word combinatorics.
    Technical translation device (Def. 3.8); standard in the subfield once introduced.

how reviews work

0 comments
Cite this review

Pith. "Pith review of $\Pi^0_4$ conservation of a Carlson-Simpson lemma for 1-variable words." pith.science (2026). https://pith.science/paper/5XNL4NQZ

@misc{pith2026260728116,
  author       = {Pith},
  title        = {Pith review of: $\Pi^0_4$ conservation of a Carlson-Simpson lemma for 1-variable words},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5XNL4NQZ}},
  note         = {Machine review of arXiv:2607.28116}
}
abstract

Carlson and Simpson proved that for every finite coloring of the 1-variable words over a finite alphabet~$A$, there is an infinite $\omega$-variable word on which all the 1-variable words are monochromatic. This statement for $\ell$-colorings, written $\mathsf{CSL}^1_\ell$, is known to be strictly weaker than $\mathsf{ACA}_0$. We prove that $\mathsf{RCA}_0 + \mathsf{CSL}^1_2$ is a $\forall \Pi^0_4$-conservative extension of $\mathsf{RCA}_0 + \mathsf{B}Sigma_2$. Among its consequences, it implies that neither the indivisibility of the universal triangle-free Henson graph for 2-colorings, nor the tree theorem for pairs and two colors, imply $\Sigma^0_2$-induction. This answers a question of Chong, Li, Wang and Yang.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

55 extracted references · 4 linked inside Pith

  1. [1]

    Carlson-Simpson’s lemma and applications in reverse mathematics.Ann

    Paul-Elliot Angles d’Auriac, Lu Liu, Bastien Mignoty, and Ludovic Patey. Carlson-Simpson’s lemma and applications in reverse mathematics.Ann. Pure Appl. Logic, 174(9):Paper No. 103287, 16, 2023. 29

    2023

  2. [2]

    Milliken’s tree theorem and its applications: a computability-theoretic perspective.arXiv preprint arXiv:2007.09739, page 141, 2020

    Paul-Elliot Angles d’Auriac, Peter A Cholak, Damir D Dzhafarov, Benoˆıt Monin, and Ludovic Patey. Milliken’s tree theorem and its applications: a computability-theoretic perspective.arXiv preprint arXiv:2007.09739, page 141, 2020

    arXiv 2007

  3. [3]

    Baumgartner

    James E. Baumgartner. A short proof of Hindman’s theorem.Journal of Combinatorial Theory, Series A, 17(3):384–386, 1974

    1974

  4. [4]

    Blass, Jeffry L

    Andreas R. Blass, Jeffry L. Hirst, and Stephen G. Simpson. Logical analysis of some theorems of combinatorics and topological dynamics. InLogic and combinatorics (Arcata, Calif., 1985), volume 65 ofContemp. Math., pages 125–156. Amer. Math. Soc., Providence, RI, 1987

    1985

  5. [5]

    The reverse mathe- matics of Carlson’s theorem for located words.J

    Tristan Bompard, Lu Liu, and Ludovic Levy Patey. The reverse mathe- matics of Carlson’s theorem for located words.J. Comb., 16(2):251–279, 2025

    2025

  6. [6]

    Timothy J. Carlson. Some unifying principles in Ramsey theory.Discrete Math., 68(2-3):117–169, 1988

    1988

  7. [7]

    Carlson and Stephen G

    Timothy J. Carlson and Stephen G. Simpson. A dual form of Ramsey’s theorem.Adv. in Math., 53(3):265–290, 1984

    1984

  8. [8]

    The henson graphs: colorings and codings, 2026

    Peter Cholak, Natasha Dobrinen, and Charlie McCoy. The henson graphs: colorings and codings, 2026

    2026

Show all 55 references
  1. [9]

    The finite big ram- sey degrees of henson graphs are provable in ACA 0, 2026

    Peter Cholak, Natasha Dobrinen, and Henry Towsner. The finite big ram- sey degrees of henson graphs are provable in ACA 0, 2026

    2026

  2. [10]

    C. T. Chong, Wei Li, Wei Wang, and Yue Yang. On the strength of Ram- sey’s theorem for trees.Adv. Math., 369:107180, 39, 2020

    2020

  3. [13]

    The Strength of Ramsey’s Theorem For Pairs over trees: IV

    Chi Tat Chong, Wei Li, Lu Liu, and Yue Yang. The Strength of Ramsey’s Theorem For Pairs over trees: IV. Erdos-Rado’s theorem for rationals

  4. [14]

    Conservation strength of the infinite pigeonhole principle for trees.Israel Journal of Mathematics, pages 1–24, 2023

    Chi Tat Chong, Wei Wang, and Yue Yang. Conservation strength of the infinite pigeonhole principle for trees.Israel Journal of Mathematics, pages 1–24, 2023

    2023

  5. [15]

    Hirst, and Timothy H

    Jennifer Chubb, Jeffry L. Hirst, and Timothy H. McNicholl. Reverse mathe- matics, computability, and partitions of trees.J. Symbolic Logic, 74(1):201– 215, 2009. 30

    2009

  6. [16]

    Groszek, and Joseph R

    Jared Corduan, Marcia J. Groszek, and Joseph R. Mileti. Reverse mathe- matics and Ramsey’s property for trees.J. Symbolic Logic, 75(3):945–954, 2010

    2010

  7. [17]

    ProQuest LLC, Ann Arbor, MI, 1980

    Denis Campau Devlin.SOME PARTITION THEOREMS AND ULTRA- FILTERS ON OMEGA. ProQuest LLC, Ann Arbor, MI, 1980. Thesis (Ph.D.)–Dartmouth College

    1980

  8. [18]

    The Ramsey theory of the universal homogeneous triangle-free graph.Journal of Mathematical Logic, 20(2):2050012, 75,

    Natasha Dobrinen. The Ramsey theory of the universal homogeneous triangle-free graph.Journal of Mathematical Logic, 20(2):2050012, 75,

  9. [19]

    American Mathematical Society, Providence, RI, 2016

    Pandelis Dodos and Vassilis Kanellopoulos.Ramsey theory for product spaces, volume 212 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2016

    2016

  10. [20]

    A den- sity version of the Carlson-Simpson theorem.J

    Pandelis Dodos, Vassilis Kanellopoulos, and Konstantinos Tyros. A den- sity version of the Carlson-Simpson theorem.J. Eur. Math. Soc. (JEMS), 16(10):2097–2164, 2014

  11. [21]

    Ef- fectiveness for the dual ramsey theorem.Notre Dame Journal of Formal Logic, 62(3):455–490, 2021

    Damir Dzhafarov, Stephen Flood, Reed Solomon, and Linda Westrick. Ef- fectiveness for the dual ramsey theorem.Notre Dame Journal of Formal Logic, 62(3):455–490, 2021

    2021

  12. [22]

    Dzhafarov and Carl Mummert.Reverse mathematics—problems, reductions, and proofs

    Damir D. Dzhafarov and Carl Mummert.Reverse mathematics—problems, reductions, and proofs. Theory and Applications of Computability. Springer, Cham, [2022]©2022

    2022

  13. [23]

    Dzhafarov and Ludovic Patey

    Damir D. Dzhafarov and Ludovic Patey. Coloring trees in reverse mathe- matics.Adv. Math., 318:497–514, 2017

    2017

  14. [24]

    Erd˝ os and G

    P. Erd˝ os and G. Szekeres. A combinatorial problem in geometry.Compo- sitio Math., 2:463–470, 1935

    1935

  15. [25]

    Personal communication to L

    Harvey Friedman. Personal communication to L. Harrington, 1977

    1977

  16. [26]

    Graham, Bruce L

    Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer.Ramsey theory. Wiley Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc., Hoboken, NJ, 2013. Paperback edition of the second (1990) edition [MR1044995]

    2013

  17. [27]

    Perspectives in Mathematical Logic

    Petr H´ ajek and Pavel Pudl´ ak.Metamathematics of first-order arithmetic. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1998. Second printing

    1998

  18. [28]

    Finite sums from sequences within cells of a partition ofN

    Neil Hindman. Finite sums from sequences within cells of a partition ofN. J. Combinatorial Theory Ser. A, 17:1–11, 1974. 31

    1974

  19. [29]

    De Gruyter textbook

    Neil Hindman and Dona Strauss.Algebra in the Stone- ˇCech compactifica- tion: theory and applications. De Gruyter textbook. De Gruyter, Berlin ; Boston, 2nd revised and extended ed edition, 2012

    2012

  20. [30]

    ProQuest LLC, Ann Arbor, MI, 1987

    Jeffry Lynn Hirst.COMBINATORICS IN SUBSYSTEMS OF SECOND ORDER ARITHMETIC. ProQuest LLC, Ann Arbor, MI, 1987. Thesis (Ph.D.)–The Pennsylvania State University

    1987

  21. [31]

    Π 0 4 conservation of the Ordered Variable Word theorem.arXiv preprint arXiv:2404.18749, 2024

    Quentin Le Hou´ erou and Ludovic Levy Patey. Π 0 4 conservation of the Ordered Variable Word theorem.arXiv preprint arXiv:2404.18749, 2024

    2024 arXiv

  22. [32]

    Largeness notions and polytime translation for@Σ 0 3-consequences ofRT 2 2, 2026

    Quentin Le Hou´ erou and Ludovic Patey. Largeness notions and polytime translation for@Σ 0 3-consequences ofRT 2 2, 2026

    2026

  23. [33]

    Big ramsey degrees using parameter spaces.arXiv preprint arXiv:2009.00967, page 20, 2020

    Jan Hubiˇ cka. Big ramsey degrees using parameter spaces.arXiv preprint arXiv:2009.00967, page 20, 2020

    2009 arXiv

  24. [34]

    The Clarendon Press, Oxford University Press, New York, 1991

    Richard Kaye.Models of Peano arithmetic, volume 15 ofOxford Logic Guides. The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications

    1991

  25. [35]

    Rapidly growing Ramsey functions

    Jussi Ketonen and Robert Solovay. Rapidly growing Ramsey functions. Ann. of Math. (2), 113(2):267–314, 1981

    1981

  26. [36]

    L. A. S. Kirby and J. B. Paris. Initial segments of models of Peano’s axioms. InSet theory and hierarchy theory, V (Proc. Third Conf., Bierutowice, 1976), Lecture Notes in Math., Vol. 619, pages 211–226. Springer, Berlin, 1977

    1976

  27. [37]

    Ramsey’s theorem for pairs, collection, and proof size.Journal of Mathe- matical Logic, page 2350007, 2023

    Leszek Aleksander Ko lodziejczyk, Tin Lok Wong, and Keita Yokoyama. Ramsey’s theorem for pairs, collection, and proof size.Journal of Mathe- matical Logic, page 2350007, 2023

    2023

  28. [38]

    Some upper bounds on ordinal-valued Ramsey numbers for colourings of pairs.Selecta Math

    Leszek Aleksander Ko lodziejczyk and Keita Yokoyama. Some upper bounds on ordinal-valued Ramsey numbers for colourings of pairs.Selecta Math. (N.S.), 26(4):Paper No. 56, 18, 2020

    2020

  29. [39]

    Π 0 4 con- servation of Ramsey’s theorem for pairs

    Quentin Le Hou´ erou, Ludovic Levy Patey, and Keita Yokoyama. Π 0 4 con- servation of Ramsey’s theorem for pairs. In preparation

  30. [40]

    Corrigendum to: Π 0 4 conservation of Ramsey’s theorem for pairs

    Quentin Le Hou´ erou, Ludovic Patey, and Keita Yokoyama. Corrigendum to: Π 0 4 conservation of Ramsey’s theorem for pairs. 2026

    2026

  31. [41]

    Π0 4 conser- vation of ramsey’s theorem for pairs.Journal of the London Mathematical Society, 113(1):e70419, 2026

    Quentin Le Hou´ erou, Ludovic Levy Patey, and Keita Yokoyama. Π0 4 conser- vation of ramsey’s theorem for pairs.Journal of the London Mathematical Society, 113(1):e70419, 2026

    2026

  32. [42]

    A computable analysis of vari- able words theorems.Proc

    Lu Liu, Benoit Monin, and Ludovic Patey. A computable analysis of vari- able words theorems.Proc. Amer. Math. Soc., 147(2):823–834, 2019. 32

    2019

  33. [43]

    The reverse mathematics of the ordered variable word theorem

    Lu Liu and Ludovic Patey. The reverse mathematics of the ordered variable word theorem. June 2026

    2026

  34. [44]

    Miller and Reed Solomon

    Joseph S. Miller and Reed Solomon. Effectiveness for infinite variable words and the dual Ramsey theorem.Arch. Math. Logic, 43(4):543–555, 2004

    2004

  35. [45]

    Milliken

    Keith R. Milliken. A Ramsey theorem for trees.J. Combin. Theory Ser. A, 26(3):215–237, 1979

    1979

  36. [46]

    The strength of the tree theorem for pairs in reverse math- ematics.J

    Ludovic Patey. The strength of the tree theorem for pairs in reverse math- ematics.J. Symb. Log., 81(4):1481–1499, 2016

    2016

  37. [47]

    The proof-theoretic strength of Ram- sey’s theorem for pairs and two colors.Adv

    Ludovic Patey and Keita Yokoyama. The proof-theoretic strength of Ram- sey’s theorem for pairs and two colors.Adv. Math., 330:1034–1070, 2018

    2018

  38. [48]

    N. W. Sauer. Coloring subgraphs of the Rado graph.Combinatorica, 26(2):231–253, 2006

    2006

  39. [49]

    Algebras of sets binumerable in complete extensions of arith- metic

    Dana Scott. Algebras of sets binumerable in complete extensions of arith- metic. InProc. Sympos. Pure Math, volume 5, pages 117–121, 1962

    1962

  40. [50]

    Primitive recursive bounds for van der Waerden numbers

    Saharon Shelah. Primitive recursive bounds for van der Waerden numbers. J. Amer. Math. Soc., 1(3):683–697, 1988

    1988

  41. [51]

    Simpson.Subsystems of second order arithmetic

    Stephen G. Simpson.Subsystems of second order arithmetic. Perspectives in Logic. Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie, NY, second edition, 2009

    2009

  42. [52]

    Princeton University Press, Princeton, NJ, 2010

    Stevo Todorcevic.Introduction to Ramsey spaces, volume 174 ofAnnals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2010

    2010

  43. [53]

    Hindman’s theorem: an ultrafilter argument in second order arithmetic.J

    Henry Towsner. Hindman’s theorem: an ultrafilter argument in second order arithmetic.J. Symbolic Logic, 76(1):353–360, 2011

    2011

  44. [54]

    A simple proof and some difficult examples for Hindman’s theorem.Notre Dame J

    Henry Towsner. A simple proof and some difficult examples for Hindman’s theorem.Notre Dame J. Form. Log., 53(1):53–65, 2012

    2012

  45. [55]

    Erd˝ os-moser andIΣ 2.Israel J

    Henry Towsner and Keita Yokoyama. Erd˝ os-moser andIΣ 2.Israel J. Math., 263(2):843–870, 2024. 33

    2024

  46. [2019]

    eprint: arXiv:1912.09049

    1912 arXiv

  47. [2020]

    tex.fjournal: Journal of Mathematical Logic tex.mrclass: 03E02 (03C15 03E05 03E40 03E75 05C05 05C55) tex.mrnumber: 4128725 tex.mrreviewer: G.Cherlin

Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.