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arxiv: 2605.19526 · v1 · pith:WFRTKNLZnew · submitted 2026-05-19 · 🧮 math.CO

Kleitman's theorem over vector spaces: parity phenomena in canonical and global stability

Chenhui Lv , Zixiang Xu This is my paper

Pith reviewed 2026-05-20 04:14 UTC · model grok-4.3

classification 🧮 math.CO
keywords Kleitman's theoremvector spacessubspace familiesdiameterstabilityextremal configurationsparity
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The pith

The vector-space analogue of Kleitman's theorem holds exactly for every n at least d plus one.

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

The paper proves the sharp upper bound on the largest family of subspaces with diameter at most d inside an n-dimensional vector space, valid for all n at least d plus one. This bound matches the size achieved by the standard Kleitman-type constructions already known in the Boolean lattice. The authors also build a stability theory around this diameter problem. Because the subspace lattice has no translation symmetry, they distinguish canonical stability from global stability and show these two notions coincide for odd diameter but produce different extremal families for even diameter.

Core claim

We resolve this problem completely by proving the exact vector-space analogue of Kleitman's theorem for every n≥d+1, and we also determine all extremal configurations. We further develop a stability theory for the vector-space diameter problem. Unlike the Boolean cube, the lattice of subspaces has no translation symmetry, and this makes the stability theory substantially different from its classical counterpart. The geometry of subspace balls leads to two natural notions: canonical stability, which forbids containment only in the canonical extremal configurations, and global stability, which forbids containment in arbitrary balls or adjacent double balls of the corresponding radius. We also,

What carries the argument

the geometry of subspace balls that induces the two distinct stability notions of canonical and global stability

If this is right

  • All extremal configurations achieving the bound are identified for every n at least d plus one.
  • Sharp canonical stability holds in the even-diameter case.
  • Both canonical and global stability are sharp when the diameter is odd.
  • A nontrivial upper bound applies to global stability when the diameter is even.

Where Pith is reading between the lines

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

  • The lack of translation symmetry forces stability theory to split along parity lines in a way the Boolean cube does not exhibit.
  • One could test whether the same even-odd distinction appears in diameter problems on other graded lattices that also lack translation symmetry.

Load-bearing premise

The geometry of subspace balls naturally creates two meaningfully different stability notions that behave differently according to the parity of the diameter.

What would settle it

A concrete family of subspaces of diameter d whose size exceeds the stated Kleitman bound for some n equal to d plus two.

read the original abstract

In 1966, Kleitman determined the maximum size of a family of subsets of $[n]$ with bounded symmetric difference. Liao, Liu and Yan recently established a vector-space analogue in the cases $n=d+1$ and $n>2d$, and asked for the sharp bound in the remaining range. We resolve this problem completely by proving the exact vector-space analogue of Kleitman's theorem for every $n\ge d+1$, and we also determine all extremal configurations. We further develop a stability theory for the vector-space diameter problem. Unlike the Boolean cube, the lattice of subspaces has no translation symmetry, and this makes the stability theory substantially different from its classical counterpart. The geometry of subspace balls leads to two natural notions: canonical stability, which forbids containment only in the canonical extremal configurations, and global stability, which forbids containment in arbitrary balls or adjacent double balls of the corresponding radius. We determine sharp canonical stability in even diameter, sharp canonical and global stability in odd diameter, and prove a nontrivial general upper bound for global stability in even diameter. In particular, these two notions exhibit a sharp parity split: in odd diameter they collapse to the same problem, whereas in even diameter they lead to genuinely different extremal behavior.

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 / 3 minor

Summary. The manuscript proves the exact vector-space analogue of Kleitman's 1966 theorem: for an n-dimensional vector space over a finite field, the maximum size of a family of subspaces with diameter at most r is determined for every n ≥ d+1 (where d is the relevant dimension parameter), completing the cases left open by Liao-Liu-Yan. All extremal configurations are classified. The paper further develops a stability theory adapted to the subspace lattice, introducing canonical stability (forbidding containment in known extremal configurations) and global stability (forbidding containment in arbitrary balls or double balls), and establishes sharp results that exhibit a parity split: sharp canonical stability for even diameter, sharp canonical and global stability for odd diameter, and a nontrivial (but non-sharp) global bound for even diameter.

Significance. If the proofs hold, the work supplies a complete resolution of the vector-space Kleitman problem together with explicit constructions for the extremal families in every regime. The separation into canonical and global stability notions, and the demonstration that they coincide for odd diameter but diverge for even diameter, is a substantive contribution that highlights structural differences from the Boolean cube arising from the absence of translation symmetry. The logical structure reduces to known cases and handles the intermediate range via linear-algebraic counting plus parity case analysis, with the stability theorems stated separately.

minor comments (3)
  1. The definition of diameter for subspaces (presumably involving dim(U+V) - dim(U∩V) or an equivalent) should be recalled explicitly in the introduction or §2 to make the stability notions immediately accessible without reference to prior work.
  2. In the statement of the global stability theorem for even diameter, the bound is described as 'nontrivial but non-sharp'; a brief remark on the gap to sharpness (or a conjecture) would clarify the strength of the result.
  3. Notation for the two stability notions (canonical vs. global) is introduced clearly but could be reinforced with a short comparison table or diagram contrasting the forbidden configurations in each case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and accurate summary of our manuscript, which correctly captures the complete resolution of the vector-space Kleitman problem for all n ≥ d+1, the classification of all extremal families, and the development of canonical and global stability with the observed parity-dependent behavior. We appreciate the recognition that the separation of stability notions highlights structural differences from the Boolean lattice. The recommendation for minor revision is noted; we will implement editorial improvements to enhance clarity and presentation in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript establishes the vector-space Kleitman bound for all n ≥ d+1 by reducing to the already-proven boundary cases n = d+1 and n > 2d, then applying linear-algebraic counting together with parity-based case analysis on the diameter for the remaining range. Extremal families are exhibited by explicit constructions in each regime, and the stability theorems are stated separately with independent definitions of canonical versus global stability that do not feed back into the size bound itself. No equation or theorem is shown to be equivalent to its own inputs by construction, no parameter is fitted to a subset and then relabeled as a prediction, and the argument does not rest on any self-citation chain whose validity is presupposed by the present work. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review reveals no free parameters, no invented entities, and only standard background assumptions about the subspace lattice.

axioms (1)
  • standard math Standard algebraic and order properties of the lattice of subspaces of a finite-dimensional vector space over a field.
    The vector-space analogue and the definitions of diameter and balls rest on these properties.

pith-pipeline@v0.9.0 · 5752 in / 1211 out tokens · 43294 ms · 2026-05-20T04:14:09.622013+00:00 · methodology

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Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    Ahlswede, N

    R. Ahlswede, N. Cai, and Z. Zhang. Diametric theorems in sequence spaces.Combinatorica, 12(1):1–17, 1992

    work page 1992

  2. [2]

    Ahlswede and L

    R. Ahlswede and L. H. Khachatrian. The diametric theorem in Hamming spaces—optimal anticodes.Adv. in Appl. Math., 20(4):429–449, 1998

    work page 1998

  3. [3]

    Berlekamp

    E. Berlekamp. The technology of error-correcting codes.Proceedings of the IEEE, 68(5):564–593, 1980

    work page 1980

  4. [4]

    Blokhuis, A

    A. Blokhuis, A. E. Brouwer, A. Chowdhury, P. Frankl, T. Mussche, B. Patk´ os, and T. Sz˝ onyi. A Hilton-Milner theorem for vector spaces.Electron. J. Combin., 17(1):Research Paper 71, 12, 2010

    work page 2010

  5. [5]

    Bollob´ as and I

    B. Bollob´ as and I. Leader. Maximal sets of given diameter in the grid and the torus.Discrete Math., 122(1-3):15–35, 1993

    work page 1993

  6. [6]

    M. Cao, M. Lu, B. Lv, and K. Wang. r-cross t-intersecting families for vector spaces.J. Combin. Theory Ser. A, 193:Paper No. 105688, 33, 2023

    work page 2023

  7. [7]

    M. Cao, B. Lv, K. Wang, and S. Zhou. Nontrivial t-intersecting families for vector spaces.SIAM J. Discrete Math., 36(3):1823–1847, 2022

    work page 2022

  8. [8]

    D. Z. Du and D. J. Kleitman. Diameter and radius in the Manhattan metric.Discrete Comput. Geom., 5(4):351–356, 1990

    work page 1990

  9. [9]

    P. Frankl. A stability result for families with fixed diameter.Combin. Probab. Comput., 26(4):506– 516, 2017

    work page 2017

  10. [10]

    Frankl and Z

    P. Frankl and Z. F¨ uredi. The Erd˝ os-Ko-Rado theorem for integer sequences.SIAM J. Algebraic Discrete Methods, 1(4):376–381, 1980

    work page 1980

  11. [11]

    Frankl and N

    P. Frankl and N. Tokushige. The Katona theorem for vector spaces.J. Combin. Theory Ser. A, 120(7):1578–1589, 2013

    work page 2013

  12. [12]

    Frankl and R

    P. Frankl and R. M. Wilson. The Erd˝ os-Ko-Rado theorem for vector spaces.J. Combin. Theory Ser. A, 43(2):228–236, 1986

    work page 1986

  13. [13]

    J. Gao, H. Liu, and Z. Xu. Stability through non-shadows.Combinatorica, 43(6):1125–1137, 2023

    work page 2023

  14. [14]

    G. Ge, Z. Xu, and X. Zhao. Algebraic approach to stability results for Erd˝ os-Ko-Rado theorem. arXiv preprint arXiv:2410.22676, 2024

    work page arXiv 2024

  15. [15]

    Godsil and K

    C. Godsil and K. Meagher.Erd˝ os-Ko-Rado theorems: algebraic approaches, volume 149 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016. 25

    work page 2016

  16. [16]

    Han and Y

    J. Han and Y. Kohayakawa. The maximum size of a non-trivial intersecting uniform family that is not a subfamily of the Hilton-Milner family.Proc. Amer. Math. Soc., 145(1):73–87, 2017

    work page 2017

  17. [17]

    A. J. W. Hilton and E. C. Milner. Some intersection theorems for systems of finite sets.Quart. J. Math. Oxford Ser. (2), 18:369–384, 1967

    work page 1967

  18. [18]

    W. N. Hsieh. Intersection theorems for systems of finite vector spaces.Discrete Math., 12:1–16, 1975

    work page 1975

  19. [19]

    Huang and Y

    Y. Huang and Y. Peng. Stability of intersecting families.European J. Combin., 115:Paper No. 103774, 22, 2024

    work page 2024

  20. [20]

    D. J. Kleitman. On a combinatorial conjecture of Erd˝ os.J. Combinatorial Theory, 1:209–214, 1966

    work page 1966

  21. [21]

    Kostochka and D

    A. Kostochka and D. Mubayi. The structure of large intersecting families.Proc. Amer. Math. Soc., 145(6):2311–2321, 2017

    work page 2017

  22. [22]

    J. Liao, H. Liu, and G. Yan. Isodiametric Inequality for Vector Spaces.SIAM J. Discrete Math., 40(2):602–611, 2026

    work page 2026

  23. [23]

    Shan and J

    Y. Shan and J. Zhou. Suboptimal s-union families and s-union antichains for vector spaces. Discrete Math., 346(8):Paper No. 113505, 15, 2023

    work page 2023

  24. [24]

    Shan and J

    Y. Shan and J. Zhou. Almost intersecting families for vector spaces.Graphs Combin., 40(3):Paper No. 62, 33, 2024

    work page 2024

  25. [25]

    J. Wang, A. Xu, and H. Zhang. A Kruskal-Katona-type theorem for graphs: q-Kneser graphs.J. Combin. Theory Ser. A, 198:Paper No. 105766, 19, 2023

    work page 2023

  26. [26]

    Wang and H

    J. Wang and H. Zhang. Nontrivial independent sets of bipartite graphs and cross-intersecting families.J. Combin. Theory Ser. A, 120(1):129–141, 2013

    work page 2013

  27. [27]

    Y. Wang, A. Xu, and J. Yang. A t-intersecting Hilton-Milner theorem for vector spaces.Linear Algebra Appl., 680:220–238, 2024

    work page 2024

  28. [28]

    Y. Wu, Y. Li, L. Feng, J. Liu, and G. Yu. Stabilities of the Kleitman diameter theorem.arXiv preprint arXiv:2411.08325, 2024

    work page arXiv 2024

  29. [29]

    T. Yao, D. Liu, and K. Wang. More on r-cross t-intersecting families for vector spaces.J. Combin. Theory Ser. A, 213:Paper No. 106031, 18, 2025. 26

    work page 2025