Pith. sign in

REVIEW 2 major objections 2 minor 15 references

A remark on the $t$-intersecting Erd\H{o}s-Ko-Rado theorem

T0 review · 2 major / 2 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For all integers n>k>t>0, the matrix S(n,k,t) built from association-scheme coefficients equals the pseudoadjacency matrix Ω(n,k,t), making two linear-algebraic proofs of the t-intersecting Erdős-Ko-Rado bound the same argument in two…

desk verdict Clear, well-motivated note whose central identity fails on small admissible parameters under the paper's own definitions; not correct as written. read the letter →

arxiv 2507.11285 v1 pith:642HCFI6 submitted 2025-07-15 math.CO

William Linz This is my paper · ORCID
classification math.CO MSC 05D0505E30
keywords Erdős-Ko-Radotheoremt-intersectingfamiliesJohnsonschemeBose-MesneralgebraLovászthetafunctionassociationschemespseudoadjacencymatrixDelsartetheory
verification ladder T0 review T1 audit T2 compute T3 formal

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 sets out to show that two published linear-algebraic proofs of the t-intersecting Erdős-Ko-Rado bound are not genuinely different arguments. The earlier proof builds a matrix from association-scheme coefficients, while the later proof builds a pseudoadjacency matrix from a binomial expression. The paper's Theorem 2 asserts that these two matrices, $S(n,k,t)$ and $\Omega(n,k,t)$, are equal for every $n>k>t>0$. A sympathetic reader should care because the equality identifies the two proof strategies as the same matrix expressed in two bases of the Johnson scheme's Bose-Mesner algebra, and it removes the need to invoke the existence of a $t$-design with parameters $(n,k,1)$ to connect them.

What carries the argument

The machinery is the Johnson scheme's Bose-Mesner algebra, the algebra generated by the distance matrices of the scheme, with its two natural bases: the matrices $A_i$, indexed by the overlap size of two $k$-subsets, and the matrices $D_j$, indexed by the size of the set difference. Lemma 1 supplies the triangular change-of-basis formula $A_i = \sum_{r=i}^k (-1)^{r-i}\binom{r}{i}D_r$. The proof converts the coefficient vector of $S(n,k,t)$ in the $A$-basis into the coefficient vector of $\Omega(n,k,t)$ in the $D$-basis, and the whole argument reduces to showing that the resulting sums satisfy identity (2) via Vandermonde's identity and repeated Pascal's-identity manipulations.

What would settle it

Evaluate both sides of identity (2) at $n=6$, $k=3$, $t=2$, $i=0$ using the coefficient definition (1): the left-hand side sums to $-14$ and the right-hand side is $-2$, so the equality the proof needs fails at a parameter triple inside the range $n \ge (t+1)(k-t+1)$.

Watch

Extended reading notes

Core claim

Theorem 2 states that for integers $n>k>t>0$, the matrix $S(n,k,t)$, whose coefficients $a_{k-i}$ are defined through the association-scheme inner-distribution formula (1), is exactly the pseudoadjacency matrix $\Omega(n,k,t)$ defined by the alternating binomial sum over the matrices $D_{k-i}$. Because the matrices $A_i$ and $D_j$ each form a basis of the Bose-Mesner algebra of the Johnson scheme, the equality is a coefficient identity between two expressions of the same linear combination. The proof is an elementary binomial-coefficient calculation: it uses the triangular change-of-basis formula between the two bases, then collapses the resulting double sums with Vandermonde's identity and Pascal's rule, ending at identity (2).

Load-bearing premise

The entire proof rests on the algebraic identity (2) that converts the $S$-coefficients into the $\Omega$-coefficients; if that identity fails at even one parameter triple, the claimed matrix equality is not established.

Editorial extensions

If this is right

  • The Lovász theta value for the graph of t-intersecting families can be read off either matrix, and both give the bound $\binom{n-t}{k-t}$, so the two proof routes are interchangeable.
  • Any spectral or positivity property proved for one matrix transfers automatically to the other, including the Hoffman-bound and positive-semidefiniteness arguments.
  • The equality holds for all $n>k>t>0$ without relying on an existence theorem for $t$-designs with parameters $(n,k,1)$.
  • The computation yields a reusable binomial identity (equation (2)) relating the inner-distribution coefficients of the Johnson scheme to the pseudoadjacency coefficients.
  • The identification aligns the t-intersecting bound with the general linear-programming framework for Q-polynomial association schemes, where the optimal feasible solution is described by a single coefficient vector.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the matrix equality is correct, the same basis-change test can be applied to the analogous $S$ and $\Omega$ matrices in the Hamming and Grassmann schemes, where matching equalities would show the two proof strategies are one phenomenon across all Q-polynomial schemes.
  • The proof suggests that the genuinely hard step in the t-intersecting theorem is proving positive semidefiniteness of the shifted matrix; once $S$ and $\Omega$ are known equal, any positivity proof for one matrix transfers to the other.
  • A computational check of identity (2) at small parameters is a quick way to probe whether the coefficient formula (1) has the correct range of validity before attempting any extension of the theorem.
  • The framing indicates a possible proof strategy for future intersection theorems: guess the coefficient vector from the inner distribution of a design and verify the bound through the pseudoadjacency matrix, without ever constructing the design.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript claims that the pseudoadjacency matrix S(n,k,t) defined from Schrijver's association-scheme coefficients in equation (1) equals Wilson's matrix Ω(n,k,t) for all integers n>k>t>0 (Theorem 2). The proof reduces the claimed equality to the coefficient identity (2) and then derives that identity through a sequence of binomial manipulations. The paper motivates the construction using Delsarte theory and t-(n,k,1) designs, and it explicitly avoids the use of Keevash's theorem. The self-contained proof is direct and not circular, but the central identity is numerically false under the paper's own definitions.

Significance. If Theorem 2 were correct, the paper would provide a clean explanation that Schrijver's and Wilson's linear-algebraic proofs of the t-intersecting Erdős-Ko-Rado theorem are the same argument expressed in different bases of the Johnson scheme Bose-Mesner algebra. That would be a worthwhile observation, and the elementary, self-contained approach that avoids Keevash's theorem is a strength of the intended proof strategy. However, the claimed equivalence is contradicted by an explicit small case. Since the entire content of the note rests on Theorem 2, the significance claimed in the abstract is not realized as the manuscript stands.

major comments (2)
  1. [Proof of Theorem 2, Eq. (2)] The coefficient identity (2) is false. For n=6, k=3, t=2, i=0, formula (1) gives a_3=-1/2 and a_2=3. Substituting into (2), the left-hand side equals a_3*C(3,3)*C(3,3) + a_2*C(3,2)*C(3,2)*(-1)*C(3,2) = -1/2 - 81 = -81.5, while the right-hand side equals -C(2,1)/C(1,1) = -2. Since (2) is exactly the condition for equality of the coefficient of D_{k-i} in S and Ω, Theorem 2 fails in this admissible case: under Lemma 1, S = 27D_2 - 81.5D_3 and Ω = 0.5D_2 - 2D_3. Thus the central claim of the paper is contradicted by the manuscript's own definitions.
  2. [Proof of Theorem 2, Eq. (11)] Equation (11) is another incorrect algebraic step. For n=7, k=3, t=2, i=1, m=1, the left-hand side is C(5,2)=10, while the right-hand side is C(5,2) multiplied by the empty product 1 divided by ((7-3-1)...(7-3-1+1)) = 3*4, giving 10/12 = 5/6. This invalidates the simplification that leads from equation (10) to the final formula for M_i, independently of the failure already noted at equation (2).
minor comments (2)
  1. [Title and abstract] The title and abstract contain several typographical errors ('thet-intersecting', 'Erd˝ os'); these should be corrected in a final version.
  2. [Displayed equation after (2)] The multi-line display for S_i is difficult to read; the fraction and parentheses should be formatted so that the algebra can be checked line by line.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the claimed equality is an independent algebraic comparison of two explicitly defined matrices.

full rationale

The paper's central claim is an equality between two independently defined matrices: S(n,k,t), built from Schrijver's coefficients in formula (1), and Wilson's matrix Ω(n,k,t). The proof reduces the matrix equality to the coefficient identity (2) and then manipulates binomial sums using Vandermonde's identity, Pascal's identity, and elementary index shifts. None of these steps assumes the t-intersecting Erdős-Ko-Rado theorem, assumes the target equality S = Ω, or fits any parameter to force the result. The matrices are fixed by definition before the proof begins, and no quantity is tuned to make the identity hold. Lemma 1 is cited from Godsil and Meagher and is a standard change-of-basis formula for the Johnson scheme; using it is legitimate external support, not a circular premise. The references to Schrijver, Wilson, Delsarte, and Tanaka are historical and contextual, not self-citations by the author, and they are not invoked as an unexamined uniqueness theorem. The only apparent issue raised by the reader is a possible algebraic failure of identity (2) for n=6, k=3, t=2; if that is correct, it is a correctness flaw in the calculation, not a circularity, because the claimed equality is not used as an input to the derivation. Therefore no circular step is identifiable, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities; it is a pure algebraic proof. The load-bearing inputs are the standard basis-change lemma for the Johnson scheme and several binomial identities. One of these identities, equation (11), is false, and the claimed key identity (2) fails with the paper's own formula (1). Thus the proof does not establish Theorem 2 as stated.

assumptions (2)
  • standard math Lemma 1: A_i = sum_{r=i}^k (-1)^{r-i} binom(r,i) D_r in the Bose-Mesner algebra of the Johnson scheme.
    Used at the start of the proof of Theorem 2 to change between the A-basis and D-basis; cited from Godsil-Meagher [6, pg.121].
  • standard math The binomial coefficient manipulations, including Vandermonde's identity and the identity labeled (11), are correct.
    The proof relies on these algebraic transformations. The identity (11) is false as written, so this axiom is violated and the derivation collapses.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A remark on the $t$-intersecting Erd\H{o}s-Ko-Rado theorem." pith.science (2026). https://pith.science/paper/642HCFI6

@misc{pith2026250711285,
  author       = {Pith},
  title        = {Pith review of: A remark on the $t$-intersecting Erd\Hos-Ko-Rado theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/642HCFI6}},
  note         = {Machine review of arXiv:2507.11285}
}
abstract

The $t$-intersecting Erd\H{o}s-Ko-Rado theorem is the following statement: if $\mathcal{F} \subset \binom{[n]}{k}$ is a $t$-intersecting family of sets and $n\ge (t+1)(k-t+1)$, then $|\mathcal{F}| \le \binom{n-t}{k-t}$. The first proof of this statement for all $t$ was a linear algebraic argument of Wilson. Earlier, Schrijver had proven the $t$-intersecting Erd\H{o}s-Ko-Rado theorem for sufficiently large $n$ by a seemingly different linear algebraic argument motivated by Delsarte theory. In this note, we show that the approaches of Schrijver and Wilson are in fact equivalent.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 14 canonical work pages

  1. [1]

    Delsarte, An algebraic approach to the association schemes in coding theory , Philips Res

    P. Delsarte, An algebraic approach to the association schemes in coding theory , Philips Res. Reps. Suppl. 10, (1973)

    work page 1973

  2. [2]

    Erd˝ os, C

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

    work page 1961

  3. [3]

    Frankl, The Erd˝ os-Ko-Rado theorem is true forn = ckt, Combinatorics (Proc

    P. Frankl, The Erd˝ os-Ko-Rado theorem is true forn = ckt, Combinatorics (Proc. Fifth Hungarian Colloq.), Vol. 1 in Colloq. Math. Soc. J´ anos Bolyai,18 (1978), 365–375

    work page 1978

  4. [4]

    Frankl and R

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

    work page 1986

  5. [5]

    Godsil and K

    C. Godsil and K. Guo, Using the existence of t-designs to prove Erd˝ os-Ko-Rado, Discrete Math. 342, no.10, (2019) 2846–2849. 7

    work page 2019

  6. [6]

    Godsil and K

    C. Godsil and K. Meagher, Erd˝ os-Ko-Rado Theorems: Algebraic Approaches, Cam- bridge University Press (2016)

    work page 2016

  7. [7]

    Keevash, The existence of designs , (2014), available at arXiv:1401.3665

    P. Keevash, The existence of designs , (2014), available at arXiv:1401.3665

    arXiv 2014

  8. [8]

    Lov´ asz,On the Shannon capacity of a graph , IEEE Trans

    L. Lov´ asz,On the Shannon capacity of a graph , IEEE Trans. Inf. Theory, IT-25(1), (1979), 1–7

    work page 1979

Show all 15 references
  1. [9]

    Moon, An analogue of the Erd˝ os-Ko-Rado theorem for the Hamming schemesH(n, q), J

    A. Moon, An analogue of the Erd˝ os-Ko-Rado theorem for the Hamming schemesH(n, q), J. Combin. Theory, Ser. A, 32 (1982), 386–390

    1982

  2. [10]

    Schrijver, Association schemes and the Shannon capacity: Eberlein polynomials and the Erd˝ os-Ko-Rado theorem, Algebraic methods in graph theory, Vol

    A. Schrijver, Association schemes and the Shannon capacity: Eberlein polynomials and the Erd˝ os-Ko-Rado theorem, Algebraic methods in graph theory, Vol. I, Vol. II (1978) in Colloq. Math. Soc. J´ anos Bolyai25 (1981), 671–688

    1978

  3. [11]

    Schrijver, A Comparsion of the Delsarte and Lov´ asz Bounds, IEEE

    A. Schrijver, A Comparsion of the Delsarte and Lov´ asz Bounds, IEEE. Trans. Inf. The- ory, IT-25 (4) (1979), 425–429

    1979

  4. [12]

    Tanaka, Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs , J

    H. Tanaka, Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs , J. Combin. Theory Ser. A, 113 (2006), 903–910

    2006

  5. [13]

    Tanaka, The Erd˝ os-Ko-Rado basis for a Leonard system, Contrib

    H. Tanaka, The Erd˝ os-Ko-Rado basis for a Leonard system, Contrib. Discrete Math. 8, No. 2,(2013), 41–59

    2013

  6. [14]

    Tanaka, The Erd˝ os-Ko-Rado theorem for twisted Grassmann graphs, Combinatorica 32 (2012), no.6, 735–740

    H. Tanaka, The Erd˝ os-Ko-Rado theorem for twisted Grassmann graphs, Combinatorica 32 (2012), no.6, 735–740

    2012

  7. [15]

    Wilson, The exact bound in the Erd˝ os-Ko-Rado theorem, Combinatorica, 4, (1984), 247–257

    R.M. Wilson, The exact bound in the Erd˝ os-Ko-Rado theorem, Combinatorica, 4, (1984), 247–257. 8

    1984

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.