A Note on the Frankl Conjecture

Maysam Maysami Sadr

arxiv: 1907.09976 · v1 · pith:WLKNFVWRnew · submitted 2019-07-23 · 🧮 math.CO

A Note on the Frankl Conjecture

Maysam Maysami Sadr This is my paper

Pith reviewed 2026-05-24 17:20 UTC · model grok-4.3

classification 🧮 math.CO

keywords Frankl conjectureunion-closed setsunion-closed familiescombinatoricsset familiesconjecture variants

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The pith

Variants of the Frankl conjecture on union-closed sets are introduced with partial justifications.

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

This short note proposes several variants of the well-known Frankl conjecture, also called the union-closed sets conjecture. The variants are presented as modifications to the original statement about elements appearing in at least half the sets of a union-closed family. Some justifications are offered for these variants. The intent is to generate new angles on the unsolved original problem in finite set combinatorics.

Core claim

The author introduces variants of the union-closed sets conjecture and provides partial justifications for them, with the aim that these modifications yield meaningful insight into the original conjecture.

What carries the argument

Variants of the Frankl conjecture

If this is right

If a variant holds, it may restrict or guide possible counterexamples to the original conjecture.
Justifications for variants could translate into partial results or proof techniques applicable to the standard statement.
The variants might allow separate verification on restricted classes of families, such as those with bounded size.

Where Pith is reading between the lines

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

Computational checks of the variants on small random union-closed families could be performed independently to test consistency.
One or more variants might turn out to be equivalent to known results in extremal set theory, linking the conjecture to other problems.
If the variants survive scrutiny, they could motivate a search for a common generalization that includes the original as a special case.

Load-bearing premise

The variants represent non-trivial changes whose partial justifications actually illuminate the original conjecture instead of restating it trivially.

What would settle it

A counterexample showing that at least one variant is equivalent to a trivial or unrelated statement with no bearing on whether the original conjecture holds.

read the original abstract

The Frankl conjecture (called also union-closed sets conjecture) is one of the famous unsolved conjectures in combinatorics of finite sets. In this short note, we introduce and to some extent justify some variants of the Frankl conjecture.

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

A short note that floats a few variants of the Frankl conjecture with only partial justifications and no real advance on the original problem.

read the letter

This paper is a brief note that introduces some variants of the Frankl conjecture on union-closed families and claims to justify them to some extent. That is the entire contribution; there is no resolution or even a strong new reduction of the open problem. The author stays inside the usual language of finite set systems and tries to find nearby statements that might be more tractable. That is the part that is actually new, though the variants themselves are not shown in the abstract and their distance from earlier reformulations is unclear. The note does one thing cleanly: it is explicit that the support is only partial, so it does not overclaim. The writing is direct and the scope is narrow, which fits a short note format. The soft spots are straightforward. The justifications are described only as holding “to some extent,” and no concrete families, calculations, or counter-example checks appear in the supplied text, so it is impossible to judge whether the variants are non-trivial or merely cosmetic restatements. No new combinatorial identity or parameter-free argument is visible. The paper therefore adds little that a specialist could immediately use or cite. It is aimed at the small group already working on the Frankl conjecture; anyone outside that circle will find nothing actionable. The work shows honest engagement with the literature and no internal contradictions, but the evidential bar is low. I would not bring it to a reading group and would not cite it. A serious editor should desk-reject rather than send it to referees, because the claim is too modest and too lightly supported to justify referee time.

Referee Report

1 major / 0 minor

Summary. The manuscript is a short note claiming to introduce some variants of the Frankl (union-closed sets) conjecture and to justify them 'to some extent.' No specific variants, statements, or arguments appear in the provided text.

Significance. If the variants were non-trivial and the partial justifications supplied concrete new insight or a useful reformulation, the note could modestly aid work on the conjecture; however, the absence of any technical content prevents assessment of whether this holds.

major comments (1)

The abstract states that variants are introduced and justified 'to some extent' but supplies neither the variants nor the arguments, so the degree of support for the central claim cannot be verified from the manuscript.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our short note. We address the major comment below.

read point-by-point responses

Referee: The abstract states that variants are introduced and justified 'to some extent' but supplies neither the variants nor the arguments, so the degree of support for the central claim cannot be verified from the manuscript.

Authors: We agree that the submitted manuscript is extremely brief and consists only of the announcement that variants are introduced, without stating the variants themselves or providing any arguments or justifications. This makes it impossible for a reader to assess the claimed contribution from the text. We will revise the manuscript to explicitly state the variants of the Frankl conjecture and include the partial justifications. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a short note whose sole claim is the introduction of variants of the Frankl conjecture together with partial justifications. No equations, derivations, fitted parameters, or self-citations appear in the abstract or the described manuscript. The central claim is purely descriptive and does not reduce to any input by construction, self-definition, or load-bearing citation chain. The derivation chain is therefore self-contained with no circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are mentioned in the abstract.

pith-pipeline@v0.9.0 · 5540 in / 907 out tokens · 20522 ms · 2026-05-24T17:20:39.294879+00:00 · methodology

A Note on the Frankl Conjecture

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)