Extremal cross $t$-intersecting families under $t$-covering number constraints for vector spaces

Benjian Lv; Kaishun Wang; Yu Zhu

REVIEW 2 major objections 3 minor 16 references

For sufficiently large n, product-maximal cross t-intersecting pairs of subspace families with both t-covering numbers at least t+1 are exactly three explicitly described constructions.

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2026-08-05 00:49 UTC pith:P6MP56US

load-bearing objection Natural and likely-correct extension of the k=ℓ product EKR classification, but Lemma 2.4's key bound is lifted from an unpublished same-group preprint; referee should require a proof or independent verification. the 2 major comments →

arxiv 2608.00505 v1 pith:P6MP56US submitted 2026-08-01 math.CO

Extremal cross t-intersecting families under t-covering number constraints for vector spaces

Yu Zhu , Benjian Lv , Kaishun Wang This is my paper

classification math.CO MSC 05D05

keywords cross t-intersecting familiesvector spaces over finite fieldst-covering numberproduct extremal problemsubspace familiesGaussian binomial coefficientextremal combinatorics

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Two families of subspaces of dimensions k1 and k2 are cross t-intersecting when every member of one meets every member of the other in at least t dimensions. This paper asks which such pairs maximize the product of their sizes once both families are required to have t-covering number at least t+1, meaning no single t-dimensional subspace is common to all members of either family. For n at least 3k1+k2−t+5, the answer is that only three explicitly described constructions can be extremal: all subspaces meeting a fixed (t+2)-subspace in dimension at least t+1; a pair of H-families built from a t-subspace inside two hyperplanes; or a pair built from a (k2+1)-subspace and a (t+1)-subspace inside it. The proof shows that an extremal pair must have both t-covering numbers exactly t+1, and then classifies the minimal t-covers of the two families. This settles a natural product-maximum analogue of the classical extremal set problem for vector spaces over finite fields.

Core claim

The theorem (Theorem 1.1) states: if k1 ≥ k2 ≥ t+2 and n ≥ 3k1+k2−t+5, and if F1 and F2 are cross t-intersecting families of k1- and k2-subspaces with t-covering numbers at least t+1 and maximal product |F1||F2|, then the pair is one of three explicitly described constructions: all subspaces meeting a fixed (t+2)-subspace in dimension at least t+1; a pair of H-families sharing a t-subspace T and built from two hyperplanes whose intersection has dimension at least t+2; or a C1/C2 pair built from a (k2+1)-subspace M and a (t+1)-subspace L of M. The proof first uses product comparisons to rule out every case except τ_t(F1)=τ_t(F2)=t+1, then splits by whether the collections of minimal (t+1)-dim

What carries the argument

The central object is the t-covering number τ_t(F), the minimum dimension of a subspace T that meets every member of F in at least t dimensions; the constraint τ_t ≥ t+1 means no fixed t-dimensional subspace is universal for the family. Around it, the proof studies T_f and T_g, the collections of all (t+1)-dimensional t-covers of the two families. Two facts do the heavy lifting: T_f and T_g are cross t-intersecting, and a maximal t-intersecting family of (t+1)-subspaces has one of two rigid shapes, namely all such subspaces inside a fixed (t+2)-subspace or all such subspaces containing a fixed t-subspace. Together with a counting formula for subspaces of prescribed intersection type, these f

Load-bearing premise

The proof rests on two imported upper bounds on how many members of one family can avoid the minimal t-covers of the other; if either bound fails for maximal cross t-intersecting families, the product comparisons that eliminate all but the three constructions do not close.

What would settle it

For a small concrete instance (for example q=2, k1=4, k2=3, t=1, and n equal to the lower bound 3k1+k2−t+5), enumerate all cross t-intersecting pairs of subspace families with both t-covering numbers at least t+1 and compare their product sizes with the three constructions in Theorem 1.1. Any pair whose product exceeds all three constructions disproves the classification. A cheaper check targets the quoted bound in Lemma 2.4: exhibit a maximal pair for which more members of F avoid every member of T_g than the bound from [7, Proposition 2.3] allows; the rest of the proof is built on that inequ

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

The three constructions in Theorem 1.1 are the complete list of maximizers: any cross t-intersecting pair with both t-covering numbers at least t+1 and maximal product must be one of them.
Maximality forces both t-covering numbers to be exactly t+1; a pair with either number at least t+2 has product strictly smaller than the A-A construction and cannot be extremal.
Each extremal pair is built from small data — a fixed (t+2)-subspace Z, or a t-subspace T inside two hyperplanes with intersection dimension at least t+2, or a (k2+1)-subspace M with a (t+1)-subspace L of M — so the extremal families are directly constructible.
Together with the previously classified τ_t(F∪G) ≥ t+1 case, the theorem covers all nontrivial t-covering number regimes for product-maximal cross t-intersecting families of vector spaces.

Where Pith is reading between the lines

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

The Section 4 inequalities leave exponential slack in n, so the same three constructions may well remain extremal for n somewhat below the stated bound; that is an extrapolation from the proof, not a claim in the paper.
The proof's engine is the pair of minimal-cover collections T_f and T_g, so a stability version is plausible: pairs whose product is close to the maximum should be close to one of the three constructions, with the distance controlled by q and by the slack in Lemma 2.4.
The same cover-collection dichotomy is a natural template for weighted products and for r-wise cross t-intersecting families of subspaces, though those cases would require new product inequalities.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Natural and likely-correct extension of the k=ℓ product EKR classification, but Lemma 2.4's key bound is lifted from an unpublished same-group preprint; referee should require a proof or independent verification.

read the letter

Here's the honest read. This paper completes the natural product-version EKR story for vector spaces under strict t-covering constraints: for arbitrary k1 and k2, with both t-covering numbers at least t+1, the extremal cross t-intersecting families are exactly the three constructions A, H, C. That classification is genuinely new; [14] only handled k=ℓ, and [3] required at least one covering number to equal t. If the theorem holds, it's a solid endpoint for the q-analog line.

The paper does real work well. Constructions are explicit, their sizes are computed, and Section 4 is a genuine piece of q-binomial inequality pushing—concrete and checkable. The proof architecture is sound: Lemma 3.1 forces τ_t(F)=τ_t(G)=t+1, then the analysis of the (t+1)-covers T_f and T_g is organized and detailed. I found no internal contradiction.

The soft spots are real, though uneven. The main one is Lemma 2.4. It bounds the members of F that avoid every (t+1)-cover by θ_{t+1}θ²_{ℓ−t+1} binom(n−t−2,k−t−2), and the proof is a direct citation of [7, Proposition 2.3]. That bound is load-bearing: every case of Theorem 1.1 uses it, and if it fails, the product inequalities don't close. [7] is an unpublished arXiv preprint by the same research group. The citation pattern is not itself a problem, but a paper whose central proof depends on an unproved bound from an unreviewed preprint needs to fix that before acceptance—either by proving the bound or showing it is in an accepted source. Two smaller hand-waves: 'routine to check' in Lemma 2.5(iia) and 'remaining cases handled similarly' in Lemma 2.9. These are probably fillable, but they're the kind of detail that matters in a case-analysis proof.

Who is this for? People working in extremal set theory and q-analogues; it gives a clean classification. I'd bring it to a reading group and would cite it if I worked on this. It deserves a serious referee, not a desk rejection. My recommendation: send it out, and make the condition that the authors supply a complete proof of Lemma 2.4's bound (or a verifiable pointer) and fill the hand-waved steps. My guess is the theorem is right, but 'likely correct' isn't enough for a result this structural.

Referee Report

2 major / 3 minor

Summary. This paper determines the extremal pairs of cross t-intersecting families F1 subset [V/k1] and F2 subset [V/k2] over F_q in the product-size maximization problem under the constraints tau_t(F1)>=t+1 and tau_t(F2)>=t+1, for k1>=k2>=t+2 and n>=3k1+k2-t+5. Theorem 1.1 asserts that the only extremal structures are the three constructions A, H, and C. The proof introduces the collections T_f and T_g of (t+1)-dimensional t-covers and combines structural lemmas on these collections with a long list of Gaussian-binomial inequalities in Section 4. The case analysis is organized according to whether T_f and T_g are t-intersecting and according to their cardinalities.

Significance. Assuming the imported bounds are correct, this is a natural and nontrivial completion of the product-maximum classification for vector spaces under t-covering number constraints, extending the k=l case of Yao-Liu-Wang and complementing Cao-Lu-Lv-Wang. The paper gives explicit extremal constructions with exact cardinality formulas and a systematic q-binomial inequality apparatus. The main proof is long but internally consistent; I did not find a contradiction in the sections I could check. The result's value is, however, conditional on the correctness and appropriate hypotheses of the imported bound [7, Proposition 2.3], which the manuscript does not prove.

major comments (2)

[2, Lemma 2.4] The upper bound |F'| <= theta_{t+1} theta_{ell-t+1}^2 binom(n-t-2, k-t-2) is quoted from [7, Proposition 2.3] and is not stated or proved in this manuscript. This bound is the second term of g(...) and is used in every case of Theorem 1.1 through Lemmas 4.1-4.3. If [7, Prop. 2.3] has different hypotheses (e.g., it optimizes |F|+|G| rather than |F||G|, or assumes tau_t(F union G)=t+1), the product inequalities such as g(theta_{ell-t+1}-1,k,ell,t)g(theta_{k-t+1},ell,k,t) < h(k,ell,t)h(ell,k,t) need not hold. Since [7] is an unpublished same-group preprint, the manuscript should include the full statement and proof of this bound, or at least make the dependence explicit and conditional on acceptance of [7].
[3, Lemma 3.1] The proof applies [3, Lemma 2.5] to obtain (12) and (13) without stating the lemma or verifying its hypotheses for maximal cross t-intersecting families when (tau_t(F),tau_t(G)) is not (t+1,t+1). Lemma 3.1 is what forces tau_t(F1)=tau_t(F2)=t+1 in the proof of Theorem 1.1, so the exact statement of [3, Lemma 2.5] and a short verification should be included. This is less severe than the [7] issue because [3] is published, but it is still load-bearing and currently unverifiable from the text.

minor comments (3)

[2, Lemma 2.5(iia)] The sentence 'It is routine to check that F0 union F1 and G0 union G1 are cross t-intersecting' is terse: for F in F1 and G in G0, one must use that F is a hyperplane of M_f and dim(G cap M_f)>=t+1 to obtain dim(F cap G)>=t. A short justification would improve readability.
[4, equations (14)-(18)] The notation tilde c_1(x,y,t) and tilde c_2(y,t) is inconsistent with the definitions of c_1(y,t) and c_2(x,y,t); c_1 does not depend on x. Please harmonize the notation.
[General] The title and keywords use 'crosst-intersecting' without a space; please fix the typo. Also, [7] should be updated with a journal reference or a version identifier if it remains a preprint.

Circularity Check

0 steps flagged

No significant circularity: main theorem is a new product classification built on upstream structural bounds, not on the conclusion itself.

full rationale

Walking the derivation chain: Theorem 1.1 is proved by (i) Lemma 3.1, which uses [3, Lemma 2.5] to force (tau_t(F1),tau_t(F2))=(t+1,t+1); (ii) Lemma 2.4, which quotes [7, Proposition 2.3] to bound the members of F that contain no element of T_g; (iii) the Section 2 case analysis that actually proves the structural identifications (Lemmas 2.5, 2.6, 2.7, 2.8, 2.9); and (iv) the Section 4 product inequalities. The imported bounds are not restatements of Theorem 1.1: [3, Lemma 2.5] concerns the tau_t(F union G)>=t+1 case, and [7, Proposition 2.3] comes from a maximum-sum problem, not from the maximum-product problem under the present constrains. Lemma 2.4 supplies a cardinality estimate, while the classification is obtained from that estimate plus additional counting and maximality arguments. No parameter is fitted to the desired extremal configurations, and no known result is merely renamed. The paper does lean on several results by the same research group, so it is not fully self-contained; however, those results are upstream structural facts with distinct hypotheses rather than the target theorem. Should [7, Proposition 2.3] or [3, Lemma 2.5] be false or mis-stated, the proof would fail, but that is a correctness or verification risk, not circularity by construction. Overall, no step in the derivation is logically equivalent to its own input.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The theorem does not introduce free parameters or new postulated objects. It relies on standard q-binomial identities and on structural lemmas about t-covering families, mostly from published work by the same group and one arXiv preprint [7]. The main new content is the synthesis and inequality analysis, not new axioms.

axioms (5)

standard math Gaussian binomial coefficient identities and inequalities in Lemma 2.1
Used throughout Sections 2-4 to bound sizes of subspace families; standard q-calculus facts.
domain assumption Fact 1: every maximal t-intersecting family of (t+1)-subspaces is either all (t+1)-subspaces containing a fixed t-subspace or all (t+1)-subspaces inside a fixed (t+2)-subspace (Brouwer-Cohen-Neumaier [1])
Used in Lemma 2.6 and Lemma 2.8 to pin down the structure of T_f and T_g.
domain assumption Fact 2: T_f and T_g are cross t-intersecting ([3, Lemma 2.7])
Central to the case split in Section 2; if false, the classification argument has no starting point.
domain assumption Upper bounds [3, Lemma 2.5] and [7, Proposition 2.3] on sizes of families with prescribed t-covering numbers
Lemmas 2.4 and 3.1 quote these without proof; they drive the inequality comparisons.
domain assumption The families F and G are maximal cross t-intersecting, so (6) holds: all subspaces containing a minimal t-cover are in the family
Used to control F_T and G_T in Lemma 2.5 and elsewhere; maximality can be assumed since extending families cannot decrease the product.

pith-pipeline@v1.3.0-alltime-deepseek · 19647 in / 19643 out tokens · 200166 ms · 2026-08-05T00:49:02.263414+00:00 · methodology

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Cite this review

Pith. "Pith review of Extremal cross $t$-intersecting families under $t$-covering number constraints for vector spaces." pith.science (2026). https://pith.science/paper/P6MP56US

@misc{pith2026260800505,
  author       = {Pith},
  title        = {Pith review of: Extremal cross $t$-intersecting families under $t$-covering number constraints for vector spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P6MP56US}},
  note         = {Machine review of arXiv:2608.00505}
}

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read the original abstract

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$, and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F}\subseteq {V\brack k}$ and $\mathcal{G}\subseteq {V\brack \ell}$ are said to be cross $t$-intersecting if $\dim(F\cap G)\geq t$ for all $F\in\mathcal{F}$ and $G\in \mathcal{G}$. In this paper, we determine the extremal structures when $|\mathcal{F}||\mathcal{G}|$ attains the maximum value under the conditions $\dim\left(\cap_{F\in \mathcal{F}}F\right)<t$ and $\dim\left(\cap_{G\in \mathcal{G}}G\right)<t$.

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 12 canonical work pages · 1 internal anchor

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work page arXiv
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S. Suda and H. Tanaka, A cross-intersection theorem for vector spaces based on semidef- inite programming,Bull. Lond. Math. Soc.46 (2014) 342–348

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[10]

Tanaka and N

H. Tanaka and N. Tokushige, A semidefinite programming approach to cross 2- intersecting families, arXiv:2503.14844

Pith/arXiv arXiv
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Tokushige, The eigenvalue method for crosst-intersecting families,J

N. Tokushige, The eigenvalue method for crosst-intersecting families,J. Algebraic Com- bin.38 (2013) 653–662

work page 2013
[12]

K. Wang, J. Guo, F. Li, Singular linear space and its applications,Finite Fields Appl. 17 (2011), 395–406

work page 2011
[13]

Wen and B

J. Wen and B. Lv, Onr-crosst-intersecting families for vector spaces with large product of sizes,J. Combin. Theory Ser. A220 (2026) 106127

work page 2026
[14]

T. Yao, D. Liu, K. Wang, More onr-crosst-intersecting families for vector spaces,J. Combin. Theory Ser. A213 (2025) 106031

work page 2025
[15]

Zhang and B

H. Zhang and B. Wu, On a conjecture of Tokushige for cross-t-intersecting families,J. Combin. Theory Ser. B171 (2025) 49–70

work page 2025
[16]

Y. Zhu, B. Lv and K. Wang, On extremal crosst-intersecting families witht-covering number conditions, arXiv:2605.19424. 19

work page internal anchor Pith review Pith/arXiv arXiv

[1] [1]

Brouwer, A.M

A.E. Brouwer, A.M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer- Verlag, Berlin 1989

work page 1989

[2] [2]

Borg, The maximum product of weights of cross-intersecting families,J

P. Borg, The maximum product of weights of cross-intersecting families,J. London Math. Soc.(2) 94 (2016) 993–1018

2016

[3] [3]

M. Cao, M. Lu, B. Lv and K. Wang,r-crosst-intersecting families for vector spaces,J. Combin. Theory Ser.A 193 (2023) 105688

work page 2023

[4] [4]

M. Cao, M. Lu, B. Lv and K. Wang, Nearly extremal non-trivial crosst-intersecting families andr-wiset-intersecting families,European J. Combin.120 (2024) 103958

2024

[5] [5]

Frankl and J

P. Frankl and J. Wang, A product version of the Hilton-Milner-Frankl theorem,Sci. China Math.67 (2024), no. 2, 455–474

2024

[6] [6]

D. He, A. Li, B. Wu and H. Zhang, On nontrivial cross-t-intersecting families,J. Combin. Theory Ser. A217 (2026) 106095

work page 2026

[7] [7]

D. Liu, K. Wang and T. Yao, Non-trivial cross-t-intersecting families for vector spaces with the maximum sum of sizes, arXiv:2606.16637

work page arXiv

[8] [8]

Pyber, A new generalization of the Erd˝ os-Ko-Rado theorem,J

L. Pyber, A new generalization of the Erd˝ os-Ko-Rado theorem,J. Combin. Theory Ser. A43 (1986) 85–90. 18

work page 1986

[9] [9]

Suda and H

S. Suda and H. Tanaka, A cross-intersection theorem for vector spaces based on semidef- inite programming,Bull. Lond. Math. Soc.46 (2014) 342–348

work page 2014

[10] [10]

Tanaka and N

H. Tanaka and N. Tokushige, A semidefinite programming approach to cross 2- intersecting families, arXiv:2503.14844

Pith/arXiv arXiv

[11] [11]

Tokushige, The eigenvalue method for crosst-intersecting families,J

N. Tokushige, The eigenvalue method for crosst-intersecting families,J. Algebraic Com- bin.38 (2013) 653–662

work page 2013

[12] [12]

K. Wang, J. Guo, F. Li, Singular linear space and its applications,Finite Fields Appl. 17 (2011), 395–406

work page 2011

[13] [13]

Wen and B

J. Wen and B. Lv, Onr-crosst-intersecting families for vector spaces with large product of sizes,J. Combin. Theory Ser. A220 (2026) 106127

work page 2026

[14] [14]

T. Yao, D. Liu, K. Wang, More onr-crosst-intersecting families for vector spaces,J. Combin. Theory Ser. A213 (2025) 106031

work page 2025

[15] [15]

Zhang and B

H. Zhang and B. Wu, On a conjecture of Tokushige for cross-t-intersecting families,J. Combin. Theory Ser. B171 (2025) 49–70

work page 2025

[16] [16]

Y. Zhu, B. Lv and K. Wang, On extremal crosst-intersecting families witht-covering number conditions, arXiv:2605.19424. 19

work page internal anchor Pith review Pith/arXiv arXiv