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arxiv: 2606.18137 · v1 · pith:BHBR7HY5new · submitted 2026-06-16 · 🧮 math.LO

Productivity of maximal eventually different families

Lukas Schembecker This is my paper

Pith reviewed 2026-06-26 21:42 UTC · model grok-4.3

classification 🧮 math.LO
keywords maximal eventually different familiesn-productive familiesclosed familiesI0-productive familiesVan Douwen familiesforcing constructionsmaximality strengtheningsset theory of the reals
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The pith

Closed n-productive maximal eventually different families exist for every n and separate the corresponding strengthenings of maximality.

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

The paper defines an n-productive maximal eventually different family as one whose n-fold product family remains maximal. It constructs closed examples of these families for each finite n at least 1. The constructions separate the different levels of productivity-based maximality. The work further shows that stronger I0-productive families can be obtained by forcing and relates productivity to Van Douwen families.

Core claim

A maximal eventually different family is n-productive when the product family F^n is still maximal. Closed n-productive families exist that separate these strengthenings of maximality for every n greater than or equal to 1. Stronger I0-productive families can be forced and constructed, and productivity stands in a definite relation to Van Douwen families.

What carries the argument

n-productivity condition on a maximal eventually different family, under which the n-fold product family remains maximal.

If this is right

  • For each fixed n there exist closed maximal eventually different families whose n-fold product is maximal.
  • The productivity levels form a proper hierarchy of strengthenings of maximality.
  • I0-productive families are consistent with ZFC.
  • Productivity of eventually different families is directly comparable to the Van Douwen property.

Where Pith is reading between the lines

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

  • The same methods might produce families that remain productive for all finite n simultaneously.
  • The forcing techniques could transfer to other maximality notions such as maximal almost disjoint families.
  • Productivity may interact with cardinal invariants of the continuum in a controlled way.
  • One could test whether productivity implies or is implied by other combinatorial properties on functions from omega to omega.

Load-bearing premise

Closed families with the stated productivity properties can be built in ZFC or suitable forcing extensions while preserving maximality and closure.

What would settle it

An explicit ZFC proof that no closed 2-productive maximal eventually different family exists, or a forcing extension in which every maximal eventually different family fails to be 2-productive.

read the original abstract

A maximal eventually different family is called $n$-productive if the product family $\mathcal{F}^n$ is still maximal. We construct closed $n$-productive families separating these strengthenings of maximality at every $n \geq 1$. Furthermore, we show how to force and construct an even stronger type of $\mathcal{I}_0$-productive family and discuss the relation of productivity to Van Douwen families.

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

0 major / 2 minor

Summary. The manuscript constructs closed n-productive maximal eventually different families in ZFC for every n ≥ 1, separating strengthenings of maximality. It also gives a forcing construction for an I_0-productive family and discusses the relation of productivity notions to Van Douwen families.

Significance. If the constructions hold, the work supplies explicit ZFC inductive constructions and a forcing argument that separate levels of productivity for maximal eventually different families while preserving closure and maximality. The direct maintenance of the required properties at each stage of the constructions is a clear strength.

minor comments (2)
  1. [§2] The definition of the product family Φ^n and the precise meaning of 'closed' in the relevant topology should be stated explicitly in the first section where the main constructions begin.
  2. [§4] In the forcing argument for the I_0-productive family, the poset is described only at a high level; an explicit definition of the conditions and the verification that the generic preserves the productivity property would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of the results on closed n-productive maximal eventually different families and the I_0-productive forcing construction, and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; direct constructions in ZFC and forcing

full rationale

The paper presents explicit inductive constructions of closed n-productive maximal eventually different families for each n ≥ 1, together with a forcing construction for an I_0-productive family. These are carried out by direct definitions that maintain eventual difference, maximality, and closure at each stage, without reducing any claimed result to a fitted parameter, self-definition, or load-bearing self-citation. The central claims rest on verifiable properties of the constructed objects rather than on renaming or importing uniqueness from prior author work. No equations or steps reduce by construction to their own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard ZFC axioms and forcing; no free parameters or invented entities visible in abstract.

axioms (1)
  • standard math Axioms of ZFC set theory
    Standard background assumed for all constructions and forcing arguments in the field.

pith-pipeline@v0.9.1-grok · 5573 in / 1013 out tokens · 22609 ms · 2026-06-26T21:42:57.938879+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

7 extracted references · 1 canonical work pages

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    Vera Fischer and David Schrittesser. A Sacks indestructible co-analytic maximal eventually different family.Fundamenta Mathematicae, 252:179–201, 2021

    2021

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    Haim Horowitz and Saharon Shelah. A Borel maximal eventually different family.Annals of Pure and Applied Logic, 175(1, Part B):103334, 2024

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    Alexander S. Kechris.Classical Descriptive Set Theory. Springer New York, NY, 1995

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    Coanalytic families of functions

    Julia Millhouse and Lukas Schembecker. Coanalytic families of functions. https://arxiv.org/abs/2510.07222v1, 2025

    work page arXiv 2025

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    There is a Van Douwen family.Transactions of the American Mathematical Society, 362(11):5879–5891, 2010

    Dilip Raghavan. There is a Van Douwen family.Transactions of the American Mathematical Society, 362(11):5879–5891, 2010

    2010

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    Van Douwen and many non Van Douwen families.The Journal of Symbolic Logic, page 1–12, 2026

    Lukas Schembecker. Van Douwen and many non Van Douwen families.The Journal of Symbolic Logic, page 1–12, 2026

    2026

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    On Horowitz and Shelah’s maximal eventually different family.RIMS Kˆ okyˆ roku, Infinite Combinatorics and Forcing Theory, No.2042:99–105, 2016

    David Schrittesser. On Horowitz and Shelah’s maximal eventually different family.RIMS Kˆ okyˆ roku, Infinite Combinatorics and Forcing Theory, No.2042:99–105, 2016. Department of Mathematics, University of Hamburg, Bundesstraße 55, 20146 Hamburg, Germany Email address:lukas.schembecker@uni-hamburg.de

    2042