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arxiv: 2605.03828 · v2 · submitted 2026-05-05 · 🧮 math.LO

Recognition: 3 theorem links

· Lean Theorem

A Topological Rainbow Ramsey Theorem

Hannes Jakob , Jing Zhang
Authors on Pith no claims yet

Pith reviewed 2026-05-11 02:14 UTC · model grok-4.3

classification 🧮 math.LO MSC 03E3503E5503E02
keywords Ramsey theoryrainbow coloringscountable-to-one functionsclosed setslarge cardinalselementary submodelsChang's conjectureregressive functions
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The pith

It is consistent relative to suitable large cardinals that every countable-to-one coloring of pairs from the second uncountable ordinal has an injective restriction to some closed copy of the first uncountable ordinal.

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

The paper proves a relative consistency result for a rainbow-style Ramsey property at the second uncountable cardinal. Given any coloring of unordered pairs from ω₂ that assigns each color only countably often, the result guarantees a closed subset A of order type ω₁ on which all pairs receive distinct colors. This strengthens two earlier theorems on related combinatorial statements and directly answers a question posed in prior work. The argument introduces and applies several new principles built around towers of elementary submodels and games on regressive functions.

Core claim

It is consistent relative to the existence of suitable large cardinals that for any countable-to-one coloring c: [ω₂]² → ω₂, there exists a closed subset A ⊆ ω₂ of order type ω₁ such that c ↾ [A]² is injective. This statement simultaneously strengthens a theorem of Abraham, Cummings and Smyth and a theorem of Garti and Zhang while answering a question raised by Garti and Zhang.

What carries the argument

Towers of countable elementary submodels together with games on regressive functions and variants of strong Chang's conjecture, which are used to produce the required closed set A carrying the injective coloring.

If this is right

  • The new result strengthens the earlier theorem of Abraham, Cummings and Smyth on pair colorings at ω₂.
  • The new result strengthens the earlier theorem of Garti and Zhang on similar rainbow properties.
  • The new result answers the specific open question posed by Garti and Zhang.
  • The combinatorial principles developed here can be studied independently for their own consistency strength.

Where Pith is reading between the lines

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

  • The same techniques might adapt to produce analogous rainbow statements at higher cardinals once comparable principles are shown consistent.
  • Because the subset A is required to be closed, the result carries a topological flavor that could connect to questions about continuous maps or compact spaces built from ordinals.
  • Determining whether the large-cardinal hypothesis is strictly necessary would clarify the exact consistency strength of the rainbow property.

Load-bearing premise

The existence of suitable large cardinals is needed to obtain the consistency of the new combinatorial principles that drive the construction of the closed set A.

What would settle it

A concrete forcing extension or inner model containing the large cardinals in which some countable-to-one coloring c of pairs from ω₂ has no closed subset of order type ω₁ on which c is injective.

read the original abstract

We show that it is consistent relative to the existence of suitable large cardinals that for any countable-to-one coloring $c: [\omega_2]^2\to \omega_2$, there exists a closed subset $A\subseteq \omega_2$ of order type $\omega_1$ such that $c\restriction [A]^2$ is injective. This theorem simultaneously strengthens two theorems, one by Abraham, Cummings and Smyth and another one by Garti and Zhang, as well as answers a question raised by Garti and Zhang. New combinatorial principles involving towers of countable elementary submodels, games concerning regressive functions and variants of strong Chang's conjecture, which are key elements of the proof, are investigated.

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.

Referee Report

2 major / 3 minor

Summary. The paper establishes the consistency, relative to suitable large cardinals, that every countable-to-one coloring c: [ω₂]² → ω₂ admits a closed subset A ⊆ ω₂ of order type ω₁ on which c is injective. The proof introduces and deploys new combinatorial principles (towers of countable elementary submodels, games on regressive functions, and variants of strong Chang's conjecture) that are first shown to hold in a forcing extension and then used to construct the desired homogeneous set via a fusion argument preserving closedness and order type. The result simultaneously strengthens theorems of Abraham-Cummings-Smyth and Garti-Zhang while answering a question of the latter authors.

Significance. If correct, the theorem supplies a strong relative-consistency result at the interface of rainbow Ramsey theory and topological combinatorics on ordinals. The new principles appear to have independent interest beyond the present application and may be reusable for other problems involving elementary submodels or regressive functions on ω₂. The forcing construction over large cardinals is a standard and appropriate method for obtaining such consistency statements.

major comments (2)
  1. [§4] §4 (or the section defining the tower principle): the precise closure properties required of the tower of countable elementary submodels under the regressive-function game are stated only informally; it is unclear whether the game-winning strategy directly yields a closed set of order type ω₁ or whether an additional fusion step is needed, and this step is load-bearing for the main theorem.
  2. [§5] §5, the forcing construction: the preservation of the strong Chang's conjecture variant through the iteration is asserted but the book-keeping argument that ensures all countable-to-one colorings are handled is only sketched; a concrete reference to the relevant lemma or claim number is needed to verify that no coloring is omitted.
minor comments (3)
  1. [Introduction] The abstract and introduction use the phrase 'suitable large cardinals' without specifying the exact strength (e.g., supercompact or measurable); a brief parenthetical in the introduction would improve readability.
  2. [§2] Notation for the new principles (e.g., the tower and game notations) is introduced in §2 but not consistently reused in later sections; a short notation table or repeated definition would aid the reader.
  3. [Introduction] The comparison with Abraham-Cummings-Smyth and Garti-Zhang is stated in the abstract and introduction but lacks a dedicated paragraph detailing exactly which hypotheses are weakened or removed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address the major comments point by point below and will incorporate the requested clarifications in the revised version.

read point-by-point responses

  1. Referee: [§4] §4 (or the section defining the tower principle): the precise closure properties required of the tower of countable elementary submodels under the regressive-function game are stated only informally; it is unclear whether the game-winning strategy directly yields a closed set of order type ω₁ or whether an additional fusion step is needed, and this step is load-bearing for the main theorem.

    Authors: We agree that the presentation of the tower principle in Section 4 would benefit from greater formality. The definition is intended to capture that a winning strategy for the regressive-function game produces a tower of countable elementary submodels whose union is closed in the order topology and has order type ω₁, with the closure properties under the game moves ensuring this directly. The subsequent fusion argument in the section then uses this tower to build the desired homogeneous set. We will revise the text to state the required closure properties explicitly (including the precise conditions on the elementary submodels and the regressive functions) and to clarify that the game strategy itself yields the closed set of order type ω₁ without requiring a separate fusion step beyond what is already incorporated in the game. revision: yes

  2. Referee: [§5] §5, the forcing construction: the preservation of the strong Chang's conjecture variant through the iteration is asserted but the book-keeping argument that ensures all countable-to-one colorings are handled is only sketched; a concrete reference to the relevant lemma or claim number is needed to verify that no coloring is omitted.

    Authors: The referee is correct that the bookkeeping argument in the forcing construction of Section 5 is presented only in outline. The iteration is arranged so that a bookkeeping function enumerates all countable-to-one colorings from the ground model, with each coloring addressed at a stage where the forcing adds a suitable homogeneous set while preserving the variant of strong Chang's conjecture. We will expand this sketch into a more detailed explanation and add explicit cross-references to the relevant claims in the iteration (particularly those establishing the preservation and the enumeration of colorings) so that it is clear no coloring is omitted. revision: yes

Circularity Check

0 steps flagged

No circularity: relative consistency via independently verified combinatorial principles

full rationale

The paper proves a relative consistency statement by first introducing and establishing new combinatorial principles (towers of countable elementary submodels, regressive-function games, strong Chang variants) in a forcing extension from large cardinals, then deriving the rainbow Ramsey conclusion from those principles via fusion/book-keeping arguments. No load-bearing step reduces the target statement to a redefinition of its own inputs, a fitted parameter renamed as prediction, or a self-citation chain whose justification loops back to the present work; the derivation chain is self-contained and externally falsifiable against the large-cardinal assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of suitable large cardinals and on newly introduced combinatorial principles whose consistency is established in the paper.

axioms (2)
  • domain assumption Existence of suitable large cardinals
    The consistency of the rainbow statement is shown relative to them.
  • ad hoc to paper Towers of countable elementary submodels, games concerning regressive functions, and variants of strong Chang's conjecture hold in the relevant model
    These are described as key elements of the proof and are investigated as new principles.

pith-pipeline@v0.9.0 · 5404 in / 1410 out tokens · 46060 ms · 2026-05-11T02:14:41.394347+00:00 · methodology

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

  1. [1]

    Rainbow general- izations of Ramsey theory: a survey

    issn: 0012-365X. [FMO10] Shinya Fujita, Colton Magnant, and Kenta Ozeki. “Rainbow general- izations of Ramsey theory: a survey”. In: Graphs Combin. 26.1 (2010), pp. 1–30. issn: 0911-0119,1435-5914. [FMS88] M. Foreman, M. Magidor, and S. Shelah. “Martin’s Maximum, Satu- rated Ideals, and Non-Regular Ultrafilters. Part I”. In: Annals of Math- ematics 127.1 ...

    work page arXiv 2010

  2. [2]

    University of North Texas, Denton, Texas, USA, 76201 Email address : hannes.jakob@unt.edu University of North Texas, Denton, Texas, USA, 76201 Email address : jing.zhang2@unt.edu

    isbn: 0-521-38101-0. University of North Texas, Denton, Texas, USA, 76201 Email address : hannes.jakob@unt.edu University of North Texas, Denton, Texas, USA, 76201 Email address : jing.zhang2@unt.edu

    work page