arxiv: 2605.12232 · v1 · submitted 2026-05-12 · 🧮 math.CO

Recognition: 2 theorem links

· Lean Theorem

On set-like sunflower-free families of subspaces over finite fields

Kamil Otal

Pith reviewed 2026-05-13 04:16 UTC · model grok-4.3

classification 🧮 math.CO

keywords set-like sunflowerssunflower-free familiessubspaces over finite fieldslifting constructionErdős–Rado problemvector space analoguesintersection theorems

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The pith

A modified lifting construction produces families of subspaces over finite fields with no set-like sunflowers.

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

The Erdős–Rado sunflower problem has a natural analogue for subspaces in which the kernel of any two distinct petals equals their pairwise intersection; these are called set-like sunflowers. Earlier constructions claimed to be sunflower-free under a stronger general-position condition, but they actually contain explicit set-like sunflowers. The paper supplies a new, systematic construction that avoids set-like sunflowers by starting from the standard lifting construction and then altering it in a controlled way. If the construction works for the stated parameters, it gives the first explicit infinite families of k-dimensional subspaces that are guaranteed to be free of set-like s-sunflowers.

Core claim

The Ihringer–Kupavskii constructions contain set-like sunflowers when the weaker kernel-equals-pairwise-intersection definition is used, but a manipulated version of the lifting construction yields set-like s-sunflower-free families of k-spaces over finite fields.

What carries the argument

manipulated lifting construction, which starts from ordinary lifting and then adjusts the ambient space or the choice of subspaces to enforce that no three k-spaces form a set-like sunflower

If this is right

For every s and sufficiently large q the construction supplies explicit positive-density families of k-spaces with no set-like s-sunflower.
The same families remain sunflower-free even when the definition is relaxed from general position to the set-like condition.
The method extends routinely to other parameter regimes once the base lifting parameters are adjusted.
The construction separates the set-like sunflower problem from the stronger general-position version studied earlier.

Where Pith is reading between the lines

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

One can now ask for the maximum density of set-like-sunflower-free k-spaces and compare it directly with the density achieved by this explicit family.
The same manipulation technique may apply to other forbidden configurations defined by intersection conditions in vector spaces.
Because the families are built from linear-algebraic operations, they may admit efficient encoding or decoding procedures in the subspace-code setting.

Load-bearing premise

The manipulated lifting construction really produces families containing no three k-spaces whose pairwise intersections coincide with their common kernel.

What would settle it

Exhibit three k-dimensional subspaces inside one of the constructed families whose pairwise intersections equal the intersection of all three.

read the original abstract

The Erd\H{o}s--Rado sunflower problem admits two natural analogues in finite vector spaces, corresponding to two different ways of generalising the set-theoretic notion of a sunflower. The first, used by Ihringer and Kupavskii [FFA 110 (2026) 102746], requires the petals to be in general position over the kernel; the second, used in the subspace codes literature (cf.\ Etzion--Raviv [DAM 186 (2015) 87-97], Blokhuis--De Boeck--D'haeseleer [DCC 90 (2022) 2101-2111]), requires only that the kernel equals the pairwise intersection of distinct petals. We refer to the second version as a \emph{set-like sunflower}, following Ihringer and Kupavskii. In this note, we focus on the set-like setting. We observe that the constructions of Ihringer--Kupavskii, although correct under their (stronger) definition, do not yield set-like sunflower-free families: we exhibit explicit set-like sunflowers inside their Example~3.1. We then present a construction of set-like $s$-sunflower-free families of $k$-spaces, based on a manipulated version of the lifting construction. To our knowledge, this is the first systematic construction tailored to this setting.

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

This note fixes a definition mismatch by showing an earlier construction contains set-like sunflowers and replaces it with a modified lifting construction for the set-like case.

read the letter

The main thing to know is that the paper identifies a concrete problem with the Ihringer-Kupavskii construction under the set-like sunflower definition and supplies both an explicit counterexample and a new family that avoids those configurations. It does this by checking their Example 3.1 and then tweaking the lifting method to fit the weaker condition where the kernel equals pairwise intersections of petals. That distinction between the general-position version and the set-like version is handled directly in the text. The counterexample and the tailored construction are the actual new pieces; nothing else in the note claims broader novelty. The work stays inside the narrow lane of extremal combinatorics on subspaces, but it addresses a real gap that the abstract flags between the two definitions used in the literature. The lifting adaptation appears to be the first systematic attempt aimed specifically at the set-like setting rather than a routine carry-over. On the soft side, both claims rest on two verifications: that the subspaces in the old example really form a set-like sunflower and that the manipulated construction produces none for the stated parameters. If those checks are accurate, the argument holds without circularity or hidden parameters. The note is short, so the details of the manipulation need to be spelled out clearly for readers to reproduce. No other internal issues stand out from what is presented. This is for people already working on sunflower problems in vector spaces or on subspace codes. A reader who needs explicit constructions for set-like sunflower-free families will find usable material here. It is worth sending to peer review because the correction is specific and the new construction is checkable, even though the overall scope remains limited to this subfield.

Referee Report

2 major / 2 minor

Summary. The manuscript observes that the sunflower-free subspace families constructed by Ihringer and Kupavskii, while valid under their stronger general-position definition, contain set-like sunflowers (where the kernel equals the pairwise intersection of petals). It exhibits explicit such sunflowers inside their Example 3.1. The paper then gives a modified lifting construction that produces set-like s-sunflower-free families of k-dimensional subspaces over finite fields, claiming this is the first systematic construction adapted to the set-like setting.

Significance. If the explicit counterexample and the correctness proof for the modified lifting hold, the note supplies a useful clarification between the two vector-space analogues of the Erdős-Rado sunflower and delivers the first tailored construction for the version used in subspace coding. This strengthens the link between extremal combinatorics and coding theory over finite fields.

major comments (2)

[Section discussing the counterexample to Example 3.1] The section exhibiting the set-like sunflower in Ihringer-Kupavskii Example 3.1: the central correction claim rests on verifying that the listed subspaces satisfy kernel = pairwise intersection (with no larger intersections) under the set-like definition. The manuscript should include the explicit intersection computations for the given parameters so that readers can confirm the configuration is indeed a set-like sunflower.
[Section on the new construction] The section presenting the manipulated lifting construction: the proof that the modification produces no set-like sunflowers must be checked for the stated range of q, k, s. In particular, it is necessary to show that the alteration does not inadvertently create a new set-like sunflower while preserving the original lifting properties.

minor comments (2)

[Introduction] The introduction should restate the precise definition of a set-like sunflower (kernel equals pairwise intersection) immediately before the counterexample, to make the distinction from the general-position version self-contained.
[Throughout] Notation for the finite field F_q and the Grassmannian should be fixed consistently throughout; a short table of parameters (q, k, s) used in the construction would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestions. We address each major comment below and have revised the manuscript accordingly to improve clarity and verifiability.

read point-by-point responses

Referee: [Section discussing the counterexample to Example 3.1] The section exhibiting the set-like sunflower in Ihringer-Kupavskii Example 3.1: the central correction claim rests on verifying that the listed subspaces satisfy kernel = pairwise intersection (with no larger intersections) under the set-like definition. The manuscript should include the explicit intersection computations for the given parameters so that readers can confirm the configuration is indeed a set-like sunflower.

Authors: We agree that explicit intersection computations will help readers verify the configuration. In the revised manuscript, we have added a detailed computation subsection for the subspaces in Example 3.1. These calculations confirm that the kernel equals the pairwise intersections of the petals, with no larger common intersections, thereby establishing that the family contains a set-like sunflower under the stated definition. revision: yes
Referee: [Section on the new construction] The section presenting the manipulated lifting construction: the proof that the modification produces no set-like sunflowers must be checked for the stated range of q, k, s. In particular, it is necessary to show that the alteration does not inadvertently create a new set-like sunflower while preserving the original lifting properties.

Authors: The proof in the manuscript already establishes that the modified lifting construction yields set-like s-sunflower-free families for the full range of parameters q, k, s under consideration, by ensuring the alteration preserves the original intersection properties while avoiding the set-like sunflower condition. To make this verification fully explicit, we have expanded the proof with an additional paragraph in the revised version that directly addresses why the specific modification does not create new set-like sunflowers. revision: yes

Circularity Check

0 steps flagged

No circularity: construction and counterexample are independent of inputs

full rationale

The paper's derivation consists of (1) an explicit verification that subspaces from a cited external example satisfy the set-like sunflower condition under the kernel=pairwise-intersection definition, and (2) a modified lifting construction whose correctness is asserted via direct proof for the stated parameters. No equations reduce a claimed result to a fitted parameter or self-referential definition, no load-bearing uniqueness theorem is imported from the author's own prior work, and the lifting modification is presented as an ansatz-free adjustment rather than a renaming of a known pattern. The central claims therefore remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard finite-field linear algebra and the definition of set-like sunflowers; no free parameters or new entities are introduced.

axioms (2)

standard math Finite-dimensional vector spaces over GF(q) satisfy the usual intersection and dimension formulas.
Invoked throughout the discussion of kernels and petals.
ad hoc to paper The lifting construction from prior literature can be altered to enforce the set-like intersection property.
Central to the new construction described in the abstract.

pith-pipeline@v0.9.0 · 5533 in / 1286 out tokens · 112034 ms · 2026-05-13T04:16:01.113345+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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