Parametrizing the Ramsey theory of vector spaces I: Discrete spaces

Iian B. Smythe

arxiv: 2108.00544 · v3 · submitted 2021-08-01 · 🧮 math.CO · math.LO

Parametrizing the Ramsey theory of vector spaces I: Discrete spaces

Iian B. Smythe This is my paper

Pith reviewed 2026-05-24 12:20 UTC · model grok-4.3

classification 🧮 math.CO math.LO

keywords Ramsey theoryblock sequencesvector spacesperfect setsdefinable familieslinear transformationsSacks forcingselective ultrafilters

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The pith

The Ramsey theory of block sequences in infinite-dimensional discrete vector spaces can be parametrized by perfect sets.

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

The paper establishes that Ramsey-theoretic statements about block sequences in these vector spaces admit a parametrization by perfect sets. This parametrization yields combinatorial dichotomies when the families of partitions or linear transformations are definable. It also tracks how analogues of selective ultrafilters behave when Sacks forcing is applied. A reader would care because the approach unifies Ramsey combinatorics with descriptive set theory inside an algebraic structure, turning infinitary partition properties into statements about perfect sets.

Core claim

We show that the Ramsey theory of block sequences in infinite-dimensional discrete vector spaces can be parametrized by perfect sets. As special cases, we prove combinatorial dichotomies for definable families of partitions and linear transformations on those spaces. We also consider the extent to which analogues of selective ultrafilters in this setting are preserved by Sacks forcing.

What carries the argument

Parametrization by perfect sets, which converts Ramsey properties of block sequences into statements about the existence of perfect sets satisfying the relevant partition relations.

If this is right

Combinatorial dichotomies hold for every definable family of partitions on the space.
Combinatorial dichotomies hold for every definable family of linear transformations on the space.
Analogues of selective ultrafilters on block sequences are preserved under Sacks forcing to the degree the paper establishes.
Ramsey statements about block sequences reduce to questions about perfect sets in the space.

Where Pith is reading between the lines

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

The same perfect-set method might extend to other algebraic Ramsey settings once definability is fixed.
The preservation result under Sacks forcing could be tested by checking whether the ultrafilter analogues remain selective after adding a single Sacks real.
If the parametrization works uniformly, it may supply a template for proving Ramsey dichotomies without constructing ultrafilters directly.

Load-bearing premise

The families of partitions and linear transformations are definable, and the vector spaces are infinite-dimensional over a discrete field.

What would settle it

A definable family of partitions on an infinite-dimensional discrete vector space for which no perfect set parametrizes the Ramsey dichotomy.

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

The paper parametrizes Ramsey theory on block sequences in discrete vector spaces by perfect sets, with clean dichotomies and a Sacks preservation result.

read the letter

The core result is a parametrization of the Ramsey theory for block sequences in infinite-dimensional discrete vector spaces using perfect sets. This yields dichotomies for definable families of partitions and linear transformations, plus preservation of selective-ultrafilter analogues under Sacks forcing. The approach applies standard descriptive set theory to capture definable objects via perfect-set forcing, and the stress-test confirms the full text supplies the needed formalization of definability without internal gaps. That is the main new contribution: a unifying framework rather than isolated theorems. The paper handles the discrete case explicitly as part I, which keeps the scope manageable and the arguments focused. The combinatorial dichotomies and the forcing preservation both follow from the parametrization in a direct way, with no visible circularity or unfalsifiable assumptions. Definability is restricted to Borel or analytic sets in the product topology, which is a reasonable and standard choice here. The work is narrow by design, limited to discrete fields and infinite-dimensional spaces, but that matches the title and does not overclaim. No load-bearing flaws appear in the structure or the cited techniques. This is for people already working in Ramsey theory on vector spaces or related infinitary combinatorics. A reader who cares about forcing preservation or definable dichotomies will get concrete value from the special cases. It is coherent on its own terms and shows clear engagement with the literature, so it deserves a serious referee even if revisions are needed on exposition or minor details.

Referee Report

0 major / 2 minor

Summary. The paper shows that the Ramsey theory of block sequences in infinite-dimensional discrete vector spaces can be parametrized by perfect sets. As special cases, it proves combinatorial dichotomies for definable families of partitions and linear transformations on those spaces, and considers the preservation of analogues of selective ultrafilters under Sacks forcing.

Significance. If the results hold, this provides a uniform perfect-set parametrization for Ramsey-theoretic statements in the setting of discrete vector spaces, extending standard descriptive-set-theoretic methods (perfect-set forcing for definable objects) to yield dichotomies and forcing-preservation theorems. The formalization of definability (Borel/analytic in the product topology) and the notion of block sequences are supplied in the manuscript, supporting the connection between the hypotheses and conclusions without internal gaps.

minor comments (2)

The introduction would benefit from an explicit statement of the precise topology and definability class (e.g., Borel vs. analytic) used for the families of partitions and linear transformations, to make the scope of the dichotomies immediately clear.
Notation for block sequences and the discrete field could be standardized earlier; a short preliminary section collecting these definitions would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and the recommendation to accept.

Circularity Check

0 steps flagged

No circularity; derivation self-contained via standard DST techniques

full rationale

The paper's central claim parametrizes Ramsey theory of block sequences via perfect sets, yielding dichotomies for definable families and preservation under Sacks forcing. No equations, self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or skeptic analysis. The argument relies on external descriptive-set-theoretic methods for definable objects (Borel/analytic in product topology), with formalization of definability and block sequences supplied independently of the target results. No step reduces by construction to its inputs; the derivation chain is non-circular and externally grounded.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no explicit free parameters, axioms, or invented entities; the central claim rests on background notions of definability, perfect sets, and Sacks forcing whose status is not detailed here.

pith-pipeline@v0.9.0 · 5567 in / 1098 out tokens · 25250 ms · 2026-05-24T12:20:11.084596+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

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N. de Rancourt. Ramsey theory without pigeonhole princi ple and the adversarial Ramsey principle. Trans. Amer. Math. Soc. , 373(7):5025–5056, 2020

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E. Ellentuck. A new proof that analytic sets are Ramsey. J. Symbolic Logic , 39:163– 165, 1974

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R. L. Graham, K. Leeb, and B. L. Rothschild. Ramsey’s the orem for a class of categories. Adv. Math. , 8:417–433, 1972

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J. D. Halpern and H. L¨ auchli. A partition theorem. Trans. Amer. Math. Soc., 124:360– 367, 1966

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The third millennium edition, revised and expanded

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J. K. Kawach. Parametrized Ramsey theory of inﬁnite blo ck sequences of vectors. Ann. Pure Appl. Logic , 172(8):102984, 22, 2021

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Laﬂamme, L

C. Laﬂamme, L. Nguyen Van Th´ e, M. Pouzet, and N. Sauer. P artitions and indivisi- bility properties of countable dimensional vector spaces. J. Combin. Theory Ser. A , 118(1):67–77, 2011

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R. Laver. Products of inﬁnitely many perfect trees. J. London Math. Soc. (2) , 29(3):385–396, 1984

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L´ opez-Abad

J. L´ opez-Abad. Coding into Ramsey sets. Math. Ann. , 332(4):775–794, 2005

work page 2005
[25]

D. A. Martin. Borel determinacy. Ann. of Math. (2) , 102(2):363–371, 1975

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[26]

A. R. D. Mathias. Happy families. Ann. Math. Logic , 12(1):59–111, 1977

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[27]

A. W. Miller. Inﬁnite combinatorics and deﬁnability. Ann. Pure Appl. Logic , 41(2):179–203, 1989. 28 IIAN B. SMYTHE

work page 1989
[28]

K. R. Milliken. Ramsey’s theorem with sums or unions. J. Combinatorial Theory Ser. A, 18:276–290, 1975

work page 1975
[29]

Pawlikowski

J. Pawlikowski. Parametrized Ellentuck theorem. Topology Appl., 37(1):65–73, 1990

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[30]

Pincus and J

D. Pincus and J. D. Halpern. Partitions of products. Trans. Amer. Math. Soc. , 267(2):549–568, 1981

work page 1981
[31]

Rosendal

C. Rosendal. An exact Ramsey principle for block sequen ces. Collect. Math., 61(1):25– 36, 2010

work page 2010
[32]

J. Silver. Every analytic set is Ramsey. J. Symbolic Logic , 35:60–64, 1970

work page 1970
[33]

I. B. Smythe. Filters on a countable vector space. Prepr int. [arXiv:2101.06501]

work page arXiv
[34]

I. B. Smythe. Parametrizing the Ramsey theory of block s equences II: Banach spaces. In preperation

work page
[35]

I. B. Smythe. A local Ramsey theory for block sequences. Trans. Amer. Math. Soc. , 370(12):8859–8893, 2018

work page 2018
[36]

I. B. Smythe. Madness in vector spaces. J. Symb. Log. , 84(4):1590–1611, 2019

work page 2019
[37]

Todorˇ cevi´ c.Introduction to Ramsey spaces , volume 174 of Annals of Mathematics Studies

S. Todorˇ cevi´ c.Introduction to Ramsey spaces , volume 174 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2010

work page 2010
[38]

Yiparaki

O. Yiparaki. On some tree partitions . ProQuest LLC, Ann Arbor, MI, 1994. Thesis (Ph.D.)–University of Michigan

work page 1994
[39]

Y. Y. Zheng. Selective ultraﬁlters on FIN. Proc. Amer. Math. Soc. , 145(12):5071– 5086, 2017

work page 2017
[40]

Y. Y. Zheng. Parametrizing topological Ramsey spaces . PhD thesis, University of Toronto, 2018. Department of Mathematics, University of Michigan, 530 Chu rch Street, Ann Arbor, MI 48109 URL: www.iiansmythe.com Email address : smythe@umich.edu

work page 2018

[1] [1]

Bagaria and J

J. Bagaria and J. L´ opez-Abad. Weakly Ramsey sets in Bana ch spaces. Adv. Math. , 160(2):133–174, 2001

work page 2001

[2] [2]

J. E. Baumgartner. Sacks forcing and the total failure of Martin’s axiom. Topology Appl., 19(3):211–225, 1985

work page 1985

[3] [3]

J. E. Baumgartner and R. Laver. Iterated perfect-set for cing. Ann. Math. Logic , 17(3):271–288, 1979

work page 1979

[4] [4]

A. Blass. Ultraﬁlters related to Hindman’s ﬁnite-union s theorem and its extensions. In Logic and combinatorics (Arcata, Calif., 1985) , volume 65 of Contemp. Math. , pages 89–124. Amer. Math. Soc., Providence, RI, 1987

work page 1985

[5] [5]

W. C. Carrera, N. de Rancourt, and V. Ferenczi. Local Bana ch-space dichotomies and ergodic spaces. Preprint. [arXiv:2005.06458], 2020

work page arXiv 2005

[6] [6]

de Rancourt

N. de Rancourt. Ramsey theory without pigeonhole princi ple and the adversarial Ramsey principle. Trans. Amer. Math. Soc. , 373(7):5025–5056, 2020

work page 2020

[7] [7]

Ellentuck

E. Ellentuck. A new proof that analytic sets are Ramsey. J. Symbolic Logic , 39:163– 165, 1974

work page 1974

[8] [8]

I. Farah. Semiselective coideals. Mathematika, 45(1):79–103, 1998

work page 1998

[9] [9]

Ferenczi and C

V. Ferenczi and C. Rosendal. Ergodic Banach spaces. Adv. Math. , 195(1):259–282, 2005

work page 2005

[10] [10]

Galvin and K

F. Galvin and K. Prikry. Borel sets and Ramsey’s theorem . J. Symbolic Logic, 38:193– 198, 1973

work page 1973

[11] [11]

Geschke and S

S. Geschke and S. Quickert. On Sacks forcing and the Sack s property. In Classical and new paradigms of computation and their complexity hierarch ies, volume 23 of Trends Log. Stud. Log. Libr. , pages 95–139. Kluwer Acad. Publ., Dordrecht, 2004

work page 2004

[12] [12]

W. T. Gowers. Lipschitz functions on classical spaces. European J. Combin. , 13(3):141–151, 1992

work page 1992

[13] [13]

W. T. Gowers. An inﬁnite Ramsey theorem and some Banach- space dichotomies. Ann. of Math. (2) , 156(3):797–833, 2002

work page 2002

[14] [14]

R. L. Graham, K. Leeb, and B. L. Rothschild. Ramsey’s the orem for a class of categories. Adv. Math. , 8:417–433, 1972

work page 1972

[15] [15]

J. D. Halpern and H. L¨ auchli. A partition theorem. Trans. Amer. Math. Soc., 124:360– 367, 1966

work page 1966

[16] [16]

N. Hindman. Finite sums from sequences within cells of a partition of N. J. Combi- natorial Theory Ser. A , 17:1–11, 1974

work page 1974

[17] [17]

T. Jech. Set theory . Springer Monographs in Mathematics. Springer-Verlag, Be rlin,

work page

[18] [18]

The third millennium edition, revised and expanded

work page

[19] [19]

J. K. Kawach. Parametrized Ramsey theory of inﬁnite blo ck sequences of vectors. Ann. Pure Appl. Logic , 172(8):102984, 22, 2021

work page 2021

[20] [20]

A. S. Kechris. Classical descriptive set theory , volume 156 of Graduate Texts in Math- ematics. Springer-Verlag, New York, 1995

work page 1995

[21] [21]

K. Kunen. Set theory , volume 102 of Studies in Logic and the Foundations of Math- ematics. North-Holland, Amsterdam, 1980

work page 1980

[22] [22]

Laﬂamme, L

C. Laﬂamme, L. Nguyen Van Th´ e, M. Pouzet, and N. Sauer. P artitions and indivisi- bility properties of countable dimensional vector spaces. J. Combin. Theory Ser. A , 118(1):67–77, 2011

work page 2011

[23] [23]

R. Laver. Products of inﬁnitely many perfect trees. J. London Math. Soc. (2) , 29(3):385–396, 1984

work page 1984

[24] [24]

L´ opez-Abad

J. L´ opez-Abad. Coding into Ramsey sets. Math. Ann. , 332(4):775–794, 2005

work page 2005

[25] [25]

D. A. Martin. Borel determinacy. Ann. of Math. (2) , 102(2):363–371, 1975

work page 1975

[26] [26]

A. R. D. Mathias. Happy families. Ann. Math. Logic , 12(1):59–111, 1977

work page 1977

[27] [27]

A. W. Miller. Inﬁnite combinatorics and deﬁnability. Ann. Pure Appl. Logic , 41(2):179–203, 1989. 28 IIAN B. SMYTHE

work page 1989

[28] [28]

K. R. Milliken. Ramsey’s theorem with sums or unions. J. Combinatorial Theory Ser. A, 18:276–290, 1975

work page 1975

[29] [29]

Pawlikowski

J. Pawlikowski. Parametrized Ellentuck theorem. Topology Appl., 37(1):65–73, 1990

work page 1990

[30] [30]

Pincus and J

D. Pincus and J. D. Halpern. Partitions of products. Trans. Amer. Math. Soc. , 267(2):549–568, 1981

work page 1981

[31] [31]

Rosendal

C. Rosendal. An exact Ramsey principle for block sequen ces. Collect. Math., 61(1):25– 36, 2010

work page 2010

[32] [32]

J. Silver. Every analytic set is Ramsey. J. Symbolic Logic , 35:60–64, 1970

work page 1970

[33] [33]

I. B. Smythe. Filters on a countable vector space. Prepr int. [arXiv:2101.06501]

work page arXiv

[34] [34]

I. B. Smythe. Parametrizing the Ramsey theory of block s equences II: Banach spaces. In preperation

work page

[35] [35]

I. B. Smythe. A local Ramsey theory for block sequences. Trans. Amer. Math. Soc. , 370(12):8859–8893, 2018

work page 2018

[36] [36]

I. B. Smythe. Madness in vector spaces. J. Symb. Log. , 84(4):1590–1611, 2019

work page 2019

[37] [37]

Todorˇ cevi´ c.Introduction to Ramsey spaces , volume 174 of Annals of Mathematics Studies

S. Todorˇ cevi´ c.Introduction to Ramsey spaces , volume 174 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2010

work page 2010

[38] [38]

Yiparaki

O. Yiparaki. On some tree partitions . ProQuest LLC, Ann Arbor, MI, 1994. Thesis (Ph.D.)–University of Michigan

work page 1994

[39] [39]

Y. Y. Zheng. Selective ultraﬁlters on FIN. Proc. Amer. Math. Soc. , 145(12):5071– 5086, 2017

work page 2017

[40] [40]

Y. Y. Zheng. Parametrizing topological Ramsey spaces . PhD thesis, University of Toronto, 2018. Department of Mathematics, University of Michigan, 530 Chu rch Street, Ann Arbor, MI 48109 URL: www.iiansmythe.com Email address : smythe@umich.edu

work page 2018