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REVIEW 3 major objections 4 minor 37 references

Piece selection and cardinal arithmetic

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Pith's one-line read For every $\lambda\ge 2^\kappa$, the bounded ideal $I_{\kappa,\lambda}$ on $P_\kappa(\lambda)$ is not $(\lambda,2)$-distributive and its positive sets admit no countable-color almost-homogeneous partition relation.

desk verdict A likely correct negative answer to Abe's question under 2^κ≤λ, but the manuscript has a handful of real proof gaps that need patching before it is referee-ready. read the letter →

arxiv 1908.04375 v1 pith:K6TULCGT submitted 2019-08-12 math.LO

Pierre Matet This is my paper · ORCID
classification math.LO MSC 03E0503E0203E0403E55
keywords pieceselectioncoveringnumbersdistributiveidealpartitionrelationsPκ(λ)mildineffabilitytreepropertycardinalarithmetic
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 proves that once $\lambda\ge 2^\kappa$, the ideal $I_{\kappa,\lambda}$ of subsets of $P_\kappa(\lambda)=\{a\subseteq\lambda:|a|<\kappa\}$ that are bounded in the $\subseteq$-ordering lacks two natural large-cardinal-type properties. Its main theorem states that for every club $D\subseteq P_\kappa(\lambda)$, the restriction $J=I_{\kappa,\lambda}|D$ is not $(\lambda,2)$-distributive and the weak partition relation $J^+\,J\not\rightarrow (J^+)^2_\omega$ fails, so in particular $I_{\kappa,\lambda}$ itself has both failures. These failures answer two open questions in the area, showing that the bounded ideal has no countable-color almost-monochromatic large set once $2^\kappa\le\lambda$. The result converts a purely cardinal-arithmetic assumption into structural information about the Boolean algebra $P(P_\kappa(\lambda))/I_{\kappa,\lambda}$.

What carries the argument

The load-bearing machinery is the family of piece-selection principles $PS^+(\tau,\kappa,\lambda)$, $PS^*(\tau,\kappa,\lambda)$ and $PS(\tau,\kappa,\lambda)$, which ask that from each partition of $P_\kappa(\lambda)$ into fewer than $\tau$ pieces one can choose a piece so that chosen pieces have pairwise nonempty intersections above any prescribed set. These principles convert cardinal-arithmetic information—covering numbers $\mathrm{cov}(\lambda,\kappa^+,\kappa^+,\kappa)$ and the dominating number $d_\kappa$—into failures of distributivity and weak partition relations. Observation 8.1 is a key bridge, linking $\lambda^{<\kappa}$-distributivity of a fine ideal to the same weak partition relations, and the two cofinality cases are closed by Proposition 8.4 and Proposition 9.6.

What would settle it

Build a model of ZFC with $2^\kappa\le\lambda$ and $\mathrm{cf}(\lambda)\ne\kappa$ in which $I_{\kappa,\lambda}$ is $(\lambda,2)$-distributive, or in which $I^+_{\kappa,\lambda}\rightarrow (I^+_{\kappa,\lambda})^2_\omega$ holds; alternatively, show that for some completely ineffable $\kappa$ the smallest normal $(\kappa,2)$-distributive fine ideal on $\kappa$ has no stationary set, which would remove the only known input for the $\mathrm{cf}(\lambda)\ne\kappa$ case.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem 1.2: suppose $2^\kappa\le\lambda$ and let $D$ be a club subset of $P_\kappa(\lambda)$; write $J=I_{\kappa,\lambda}|D$. Then (i) $J^+\,J\not\rightarrow (J^+)^2_\omega$, (ii) $J^+\,J\not\rightarrow (J^+)^3_2$, and (iii) $J$ is not $(\lambda,2)$-distributive. Taking $D=P_\kappa(\lambda)$ gives the abstract's assertions about $I_{\kappa,\lambda}$ itself. The proof splits at the cofinality of $\lambda$: when $\mathrm{cf}(\lambda)\ne\kappa$, the argument runs through piece-selection principles, covering numbers, and stationary subsets of $\kappa$ chosen from the smallest normal $(\kappa,2)$-distributive fine ideal; when $\mathrm{cf}(\lambda)=\kappa$, a separate path uses a weak square-bracket partition failure plus the cardinal arithmetic that distributivity would force.

Load-bearing premise

When the cofinality of $\lambda$ is not $\kappa$, the proof assumes that the smallest normal $(\kappa,2)$-distributive fine ideal on $\kappa$ contains a stationary set; the paper does not establish this, and without such an $X$ the non-distributivity argument for that case is vacuous.

Editorial extensions

If this is right

  • The abstract's two consequences hold whenever $\lambda\ge 2^\kappa$: $I_{\kappa,\lambda}$ is not $(\lambda,2)$-distributive and $I^+_{\kappa,\lambda}\rightarrow (I^+_{\kappa,\lambda})^2_\omega$ fails.
  • The failure is club-stable: every restriction $J=I_{\kappa,\lambda}|D$ to a club $D$ inherits both failures and also fails the three-color version $J^+\,J\not\rightarrow (J^+)^3_2$.
  • Assuming the hypothesis SSH, the paper obtains stronger failures—for example square-bracket relations with $\lambda$ colors—together with some positive partition relations for the same ideals, depending on the cofinality of $\lambda$.
  • The case $\mathrm{cf}(\lambda)=\kappa$ is genuinely separate: the covering-number equalities used elsewhere fail there, so the theorem requires a distinct argument in that case.

Reading between the lines

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

  • The threshold $2^\kappa\le\lambda$ is likely not exact: a natural testable strengthening is whether the same non-distributivity already follows from $\lambda=\kappa^+$ under weaker cardinal-arithmetic hypotheses.
  • If the stationary-set assumption on the smallest normal $(\kappa,2)$-distributive fine ideal is the real bottleneck, the $\mathrm{cf}(\lambda)\ne\kappa$ case could be reproved by finding, for every completely ineffable $\kappa$, any normal fine ideal with a stationary set; this would make the proof independent of that particular ideal.
  • Because Observation 8.1 reduces $\lambda^{<\kappa}$-distributivity to the existence of three-color almost-homogeneous sets, failures of distributivity can be witnessed by explicit colorings; this suggests a combinatorial route for testing non-distributivity in other ideals on $P_\kappa(\lambda)$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops piece-selection principles PS, PS*, and PS+ on P_κ(λ), relates them to pcf-theoretic objects (scales, covering numbers), and applies them to questions of Abe and Usuba about the ideal I_{κ,λ}. The main advertised result, Theorem 1.2, states that if 2^κ ≤ λ and D is a club subset of P_κ(λ), then for J = I_{κ,λ}↾D the partition relations J+ J →(J+)^2_ω and J+ J →(J+)^3_2 fail and J is not (λ,2)-distributive. The proof is assembled from a network of propositions in Sections 5–9, several of which depend on the author's previous work. The abstract also asserts that I_{κ,λ} itself is not (λ,2)-distributive and that I^+_{κ,λ} → (I^+_{κ,λ})^2_ω fails.

Significance. If Theorem 1.2 is correct, it settles a natural strengthening of Abe's question under the cardinal arithmetic assumption 2^κ ≤ λ, and it gives a new connection between piece-selection principles and the non-distributivity of the ideal I_{κ,λ}. The paper is ambitious and technically rich: it introduces a useful hierarchy of selection principles, connects them to remarkably good scales and covering numbers, and separates several partition relations on P_κ(λ). The use of pcf theory, normal ideals, and the author's earlier results is appropriate. However, as written, two load-bearing steps are not fully justified: the application of PS+ in Observation 2.7 is made to a family that is not a partition in the sense of Definition 2.4, and the crucial hypothesis that there exists a stationary set in NCI_κ is never established. Because these steps feed into Theorem 1.2, the central claim is not completely verified in the present manuscript.

major comments (3)
  1. [Section 2, Observation 2.7] The proof applies PS+(τ,κ,λ) to the family Zx ∪ {Tx} for x ∈ P_κ(λ). This family is not a partition of P_κ(λ) as required by Definition 2.4: both Zx and Tx consist only of sets z with x ⊆ z, so no z with x ⊄ z belongs to any piece. The observation is then used in Proposition 3.5(ii), which feeds into Observation 6.4 and Proposition 6.19(i), and hence into Theorem 1.2(i). The gap is repairable: one can add a leftover piece Rx = {z : x ⊄ z}, apply PS+, and use the 'a ∪ b ⊆ c' clause of PS+ to rule out k(x) = Rx. But as written, the proof is invalid.
  2. [Section 8, Proposition 8.4] The hypothesis that some X ∈ NS+_κ ∩ NCI_κ exists is never proved. Since NCI_κ is the smallest normal (κ,2)-distributive fine ideal on κ, it is possible that NCI_κ = NS_κ, in which case the intersection NS+_κ ∩ NCI_κ is empty because an ideal and its positive sets are disjoint. This case occurs exactly when NS_κ itself is (κ,2)-distributive, a possibility not excluded by the paper's assumptions. If the intersection is empty, Proposition 8.4 is vacuously true but cannot be used to derive Theorem 1.2(iii) for cf(λ) ≠ κ. The manuscript needs either a proof that NS_κ is not (κ,2)-distributive (which would force NCI_κ to contain a stationary set) or a separate treatment of the case NCI_κ = NS_κ. As written, the derivation of Theorem 1.2(iii) is incomplete.
  3. [Section 6, Proposition 6.19(i)] The proof says 'By Observation 6.4, Corollary 6.10 and Proposition 6.18', but the cited results do not visibly imply the claimed failure of J+ J →(J+)^2_ω. Observation 6.4 gives cov(λ,κ+,κ+,κ) = λ only under the assumption that the positive partition relation holds. Corollary 6.10 is a square-bracket failure for λ colours, while Proposition 6.18 is a failure of the 2-colour relation J+ J →_κ (J+)^2_2. Neither of these is formally equivalent to, or obviously implied by, the existence of the ω-colour relation J+ J →(J+)^2_ω. The missing implication needs to be supplied. Since Proposition 6.19(i) is used for Theorem 1.2(i) when cf(λ) ≠ κ, this is a load-bearing gap.
minor comments (4)
  1. [Section 2, Proposition 2.15] The proof refers to 'Fact 2.14' when it should refer to Observation 2.14.
  2. [Section 6, Proposition 6.19] In the statement of Proposition 6.19 there is a typo: 'cfλ)' should be 'cf(λ)'.
  3. [Introduction, Section 1] The line 'if λ<κ = 2λ' in the discussion of Abe's result appears to be a typo; the intended expression is likely 'λ<κ = 2^λ' or 'λ<κ = 2^κ'.
  4. [Section 5, Definition 5.24] The notation d_κ and the barred version d_κ are both used in the text; the difference is sometimes obscured by the formatting, making it hard to know which cardinal is meant in Proposition 6.19 and related statements.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.2 follows from internal lemmas and independent published theorems; the self-citations are not load-bearing in a circular way.

full rationale

I walked the derivation chain for Theorem 1.2. The proof reduces to Observation 8.1 and Propositions 6.19, 8.4, and 9.6. Each of these is proved from earlier results in the paper or from published theorems. The cited self-results (Fact 6.7 and Fact 9.5 from [18]) are independent published partition-relation theorems with assumptions that do not include the target theorem; they are not fitted, renamed, or restated versions of the conclusion. Proposition 8.4's one-line proof hides an implicit, standard step: (λ,2)-distributivity of Iκ,λ|A yields the flipping property of Fact 7.4, hence mild λ-ineffability; Proposition 7.12 then gives cov(λ,κ+,κ+,κ)=λ, and Observation 8.3 gives the contradiction. This is a genuine derivation, not a definitional equivalence. The only caveat, which is a correctness/readability issue rather than circularity, is that the manuscript does not prove NS+κ∩NCIκ is nonempty (so Proposition 8.4's hypothesis could be vacuous for completely ineffable κ, and Theorem 1.2(iii) in the cf(λ)≠κ case would need an additional argument to supply a stationary X in NCIκ). This gap does not reduce the theorem to its inputs; it is an omitted support step. No parameter fitting, no renaming of known results, and no ansatz smuggled via citation occur in the derivation. The central claim is not defined in terms of the tools used to prove it.

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

No free parameters, since this is pure mathematics. The paper relies on standard ZFC, on a body of pcf theory and Pκ(λ) combinatorics from the literature (partly self-authored), and on two unproved technical assumptions in proofs: that PS+ can be applied to improper 'partitions', and that the ideal NCIκ contains a stationary set.

assumptions (4)
  • standard math ZFC standard axioms
    All theorems are set-theoretic, proven in ZFC unless specified.
  • standard math Background pcf theory results (e.g., Fact 2.10, Fact 4.4, Fact 4.7)
    Cited from Shelah's Cardinal Arithmetic and other standard sources; used throughout.
  • ad hoc to paper PS+ can be applied to partial partitions of Pκ(λ), not just genuine partitions as required by Definition 2.4
    The proof of Observation 2.7 uses families Zx∪{Tx} that do not cover Pκ(λ), violating Definition 2.4. This unsupported extension is used in later arguments.
  • domain assumption NCIκ contains a stationary set when κ is completely ineffable
    Proposition 8.4 and Proposition 6.18 require choosing X ∈ NS+_κ ∩ NCIκ. The paper does not prove the existence of such a stationary X in the smallest normal (κ,2)-distributive fine ideal.

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Pith. "Pith review of Piece selection and cardinal arithmetic." pith.science (2026). https://pith.science/paper/K6TULCGT

@misc{pith2026190804375,
  author       = {Pith},
  title        = {Pith review of: Piece selection and cardinal arithmetic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K6TULCGT}},
  note         = {Machine review of arXiv:1908.04375}
}
abstract

We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, we show that if $\lambda \geq 2^\kappa$, then (a) $I_{\kappa, \lambda}$ is not $(\lambda, 2)$-distributive, and (b) $I_{\kappa, \lambda}^+ \rightarrow (I_{\kappa, \lambda}^+)^2_\omega$ does not hold.

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