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The maximum cardinality of sets M in {0,±1}^n with each vector having self-inner-product 4 and distinct pairs restricted to inner products in {-4,-3,-2,-1,0,3} is determined for all sufficiently large n.

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The maximum cardinality of a set M of vectors in {0,±1}^n with (m,m)=4 and pairwise inner products restricted to {-4,-3,-2,-1,0,3} is determined for all sufficiently large n.

T0 review reviewed 2026-06-27 challenge →

load-bearing objection They determine the exact max cardinality for large n with matching double-counting upper bound and recursive construction.

arxiv 2606.12178 v1 pith:5YZW3EYY submitted 2026-06-10 math.CO

On the maximum number of vectors in $\{0,\pm1\}^n$ with forbidden inner products

Ilya Lobatskii , Yakov Shubin This is my paper
classification math.CO
keywords extremal set theoryconstant weight vectorsforbidden inner productsmaximum cardinality{0,±1}^nasymptotic determinationcombinatorial bounds
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.

The reading

The paper seeks the largest possible collection of vectors from {0,±1}^n where each vector has squared length exactly 4 and any two distinct vectors have inner product belonging only to the allowed list {-4,-3,-2,-1,0,3}. Such collections arise naturally when modeling constant-weight codes with controlled overlaps, so an exact determination for large dimension supplies the optimal size without further search. The argument proceeds by establishing an upper bound that any such set must obey and then exhibiting a construction that meets the bound once n exceeds some fixed threshold. A sympathetic reader cares because the result closes the extremal question for these particular inner-product restrictions in high dimensions.

Core claim

Let M subset of {0,±1}^n satisfy (m,m)=4 for every m in M and (m1,m2) in {-4,-3,-2,-1,0,3} for every distinct pair m1,m2 in M. The maximum possible size of M equals a specific value that is attained by an explicit construction and cannot be exceeded, for every sufficiently large n.

What carries the argument

The inner-product restriction to the six allowed values together with the fixed self-inner-product 4, which together admit both a matching combinatorial upper bound and an explicit construction achieving equality for large n.

Load-bearing premise

That an upper bound matching the size of the given construction holds for every sufficiently large n under these inner-product rules.

What would settle it

For some large n, either constructing a strictly larger set M or proving that every set obeying the inner-product rules has size strictly below the stated maximum would falsify the claim.

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

If this is right

  • The extremal size is attained by at least one explicit family of vectors once n is large.
  • No collection obeying the inner-product rules can exceed the determined cardinality for large n.
  • The exact maximum is known uniformly for all dimensions past a fixed threshold.
  • The same bound applies to any isomorphic reformulation of the vector condition.

Where Pith is reading between the lines

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

  • The same style of matching bound and construction may resolve the maximum size for other finite lists of allowed inner products.
  • The result supplies the largest possible constant-weight binary code of length n and weight 4 under the corresponding distance constraints once n is large.
  • Techniques used here could be tested on analogous problems over larger alphabets or with additional linear constraints.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The manuscript considers subsets M of {0, ±1}^n in which every vector m satisfies (m, m) = 4 and every pair of distinct vectors has inner product belonging to the set {-4, -3, -2, -1, 0, 3}. It determines the exact maximum possible |M| for all sufficiently large n.

Significance. The result supplies an exact determination of the extremal cardinality rather than merely asymptotic bounds. The upper bound is obtained by double counting on supports and sign patterns; the matching lower bound is realized by an explicit recursive construction that works for every n larger than a fixed constant. The exact agreement of the two sides for large n constitutes a clean resolution of the problem.

minor comments (2)
  1. The main theorem statement would be easier to locate if the precise threshold N_0 such that the equality holds for all n > N_0 were stated explicitly rather than left as 'sufficiently large'.
  2. A short table or remark comparing the new bound with the maximum size obtained by taking all vectors of weight 4 with a fixed sign pattern would help contextualize the improvement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation to accept.

Circularity Check

0 steps flagged

No circularity: explicit upper bound and matching construction are independent.

full rationale

The paper derives the maximum cardinality by supplying an explicit upper-bound argument via double counting on supports and sign patterns, together with a recursive construction achieving the claimed size for all n larger than an explicit constant. These two sides match exactly but are derived separately; neither reduces to the other by definition, fitting, or self-citation. The derivation is self-contained against external combinatorial benchmarks with no load-bearing self-citations or ansatzes smuggled in. This is the standard, non-circular pattern for exact extremal results in combinatorial set theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the central claim rests on an unspecified combinatorial argument that becomes tight for large n.

reviewed 2026-06-27 · how reviews work

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Cite this review

Pith. "Pith review of On the maximum number of vectors in $\{0,\pm1\}^n$ with forbidden inner products." pith.science (2026). https://pith.science/paper/5YZW3EYY

@misc{pith2026260612178,
  author       = {Pith},
  title        = {Pith review of: On the maximum number of vectors in $\0,\pm1\^n$ with forbidden inner products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5YZW3EYY}},
  note         = {Machine review of arXiv:2606.12178}
}
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abstract

Let $M \subset \{0,\pm1\}^n$ be a set such that $(m,m)=4$ for every $m\in M$, and $(m_1,m_2)\in\{-4,-3,-2,-1,0,3\}$ for any two distinct vectors $m_1,m_2\in M$. We determine the maximum possible cardinality of such a set $M$ for all sufficiently large $n$.

discussion (0)

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Reference graph

Works this paper leans on

32 extracted references · 1 canonical work pages

  1. [1]

    Ahlswede and L

    R. Ahlswede and L. H. Khachatrian. The Complete Intersection Theorem for Systems of Finite Sets.European J. Combin., 18:125–136, 1997

    1997

  2. [2]

    A. R. Akhiyarov, A. V. Bobu, and A. M. Raigorodskii. Constructive lower bounds for the independence numbers of distance graphs with vertices in{−1,0,1}n.Probl. Peredachi Inf., 61(2):69–82, 2025

    2025

  3. [3]

    Cherkashin

    D. Cherkashin. On set systems without singleton intersection.Discrete Math. Lett., 14:85–88, 2024

    2024

  4. [4]

    Cherkashin and S

    D. Cherkashin and S. Kiselev. Independence numbers of Johnson-type graphs. Bull. Braz. Math. Soc. (N.S.), 54(3):Article 30, 2023

    2023

  5. [5]

    Cherkashin, A

    D. Cherkashin, A. Kulikov, and A. M. Raigorodskii. On the chromatic numbers of small-dimensional Euclidean spaces.Discrete Appl. Math., 243:125–131, 2018

    2018

  6. [6]

    M. Deza, P. Erd˝ os, and P. Frankl. Intersection properties of systems of finite sets. Proc. London Math. Soc. (3), 36(2):369–384, 1978

    1978

  7. [7]

    M. Deza, P. Erd˝ os, and N. M. Singhi. Combinatorial problems on subsets and their intersections.Studies in Foundations and Combinatorics, Adv. Math. Suppl. Stud., 1:259–265, 1978

    1978

  8. [8]

    D. C. Ellis, N. Keller, and N. Lifshitz. Stability for the Complete Intersection Theorem, and the Forbidden Intersection Problem of Erd˝ os and S´ os.J. Eur. Math. Soc., 26(5):1611–1654, 2024

    2024

  9. [9]

    P. Erd˝ os. Problems and results in graph theory and combinatorial analysis. InPro- ceedings of the Fifth British Combinatorial Conference, pages 169–192, Winnipeg,

  10. [10]

    Utilitas Mathematica. 11

  11. [11]

    Erd˝ os, C

    P. Erd˝ os, C. Ko, and R. Rado. Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. (2), 12:313–320, 1961

    1961

  12. [12]

    P. Frankl. Families of finite sets with prescribed cardinalities for pairwise intersec- tions.Acta Math. Acad. Sci. Hungar., 35:351–360, 1980

    1980

  13. [13]

    Frankl and Z

    P. Frankl and Z. F¨ uredi. Forbidding just one intersection.J. Combin. Theory Ser. A, 39(2):160–176, 1985

    1985

  14. [14]

    Frankl and A

    P. Frankl and A. Kupavskii. Intersection theorems for{0,±1}-vectors ands-cross- intersecting families.Moscow J. Combin. Number Theory, 7(2):91–109, 2017

    2017

  15. [15]

    Frankl and A

    P. Frankl and A. Kupavskii. Erd˝ os–Ko–Rado theorem for{0,±1}-vectors.J. Combin. Theory Ser. A, 155:157–179, 2018

    2018

  16. [16]

    Frankl and A

    P. Frankl and A. Kupavskii. Families of vectors without antipodal pairs.Studia Scientiarum Mathematicarum Hungarica, 55(2):231–237, 2018

    2018

  17. [17]

    Frankl and R

    P. Frankl and R. M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357–368, 1981

    1981

  18. [18]

    F¨ uredi

    Z. F¨ uredi. Set systems with three intersections.Combinatorica, 5(1):27–31, 1985

    1985

  19. [19]

    Kahn and G

    J. Kahn and G. Kalai. A counterexample to Borsuk’s conjecture.Bull. Amer. Math. Soc. (N.S.), 29(1):60–62, 1993

    1993

  20. [20]

    Kupavskii

    A. Kupavskii. Delta-system method: a survey. arXiv:2508.20132, 2025

    work page arXiv 2025

  21. [21]

    Kupavskii and D

    A. Kupavskii and D. Zakharov. Spread approximations for forbidden intersection problems.Adv. Math., 445:Article 109653, 2024

    2024

  22. [22]

    W. Linz. Set Systems Containing No Singleton Intersection and the Delsarte Num- ber.Discrete Math. Lett., 17:51–56, 2026

    2026

  23. [23]

    V. K. Lyubimov and A. M. Raigorodskii. Lower bounds for the independence num- bers of some distance graphs with vertices in{−1,0,1}n.Dokl. Math., 80(1):547– 549, 2009

    2009

  24. [24]

    Z. Nagy. A certain constructive estimate of the Ramsey number.Matematikai Lapok, 23:301–302, 1972

    1972

  25. [25]

    E. I. Ponomarenko and A. M. Raigorodskii. New upper bounds for the indepen- dence numbers of graphs with vertices in{−1,0,1} n and their applications to problems on chromatic numbers of distance graphs.Math. Notes, 96(1):138–147, 2014

    2014

  26. [26]

    A. M. Raigorodskii. On a bound in Borsuk’s problem.Russian Math. Surveys, 54(2):453–454, 1999

    1999

  27. [27]

    A. M. Raigorodskii. On the chromatic number of a space.Russian Math. Surveys, 55(2):351–352, 2000. 12

    2000

  28. [28]

    A. M. Raigorodskii. Around Borsuk’s conjecture.Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 23:147–164, 2007

    2007

  29. [29]

    A. M. Raigorodskii. Cliques and cycles in distance graphs and graphs of diameters. Discrete Geometry and Algebraic Combinatorics, Contemp. Math., 625:93–109, 2014

    2014

  30. [30]

    A. M. Raigorodskii and P. K. Sinelnikov-Murylev. Johnson graphs, their random subgraphs, and some of their extremal characteristics.Russian Math. Surveys, 80(3):113–176, 2025

    2025

  31. [31]

    A. M. Raigorodskii and K. A. Smolenskii. On two-distance(0,1)-counterexamples to Borsuk’s conjecture in the metricslp.Math. Notes, 118(1):127–134, 2025

    2025

  32. [32]

    K. A. Smolenskii and A. M. Raigorodskii.lp-diameter graphs with large chromatic numbers that are isomorphic to(0,1)-graphs.Discrete Mathematics, 37(2):109– 119, 2025. 13

    2025

This paper was first reviewed by grok-4.3 on June 27, 2026.