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Refining the spread approximation extends the characterization of maximum t-intersecting permutation families to t ≤ n - n^{5/7+ε}.

2026-06-29 21:26 UTC pith:O2MD2X26

load-bearing objection They extend the structural EKR threshold for t-intersecting permutations by tightening error control in the spread approximation, reaching t up to n minus n to the 5/7 plus epsilon.

arxiv 2605.26051 v1 pith:O2MD2X26 submitted 2026-05-25 math.CO

Extremal t-intersecting Families of Permutations for Large t

Pitchayut Saengrungkongka This is my paper
classification math.CO
keywords t-intersecting familiespermutationsErdős-Ko-Rado theoremextremal combinatoricsspread approximation
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 paper proves that every maximum-sized t-intersecting family of permutations on n elements must be isomorphic to one of the A_k families for all t up to n minus n to the power 5/7 plus epsilon. This widens the range beyond Kupavskii's earlier threshold of roughly n minus n log log n over log n by improving the error control in the spread approximation method. A sympathetic reader cares because the result moves the Erdős-Ko-Rado theorem for permutations closer to its expected natural limit, where the obvious point-fixing constructions are known to be optimal.

Core claim

By refining Kupavskii's spread approximation technique, the paper proves that every t-intersecting family of permutations with maximum size must be isomorphic to A_k for some k, whenever t ≤ n - n^{5/7 + ε}.

What carries the argument

Refined spread approximation technique that controls error terms sufficiently to extend the allowable range of t.

Load-bearing premise

The spread approximation technique admits a refinement that controls the error terms sufficiently to reach the stated threshold t ≤ n - n^{5/7+ε} without introducing new post-hoc restrictions.

What would settle it

Exhibiting a single t-intersecting family of permutations on n elements that is larger than every A_k and not isomorphic to any of them, for some t = n - n^{5/7 + ε}, would disprove the claim.

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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 paper refines Kupavskii's spread approximation technique for t-intersecting families of permutations on [n]. It proves that for all t ≤ n - n^{5/7 + ε} (with ε > 0 fixed), every maximum t-intersecting family is isomorphic to one of the families A_k = {σ : σ fixes at least t + k points in {1,…,t+2k}}, extending the previous range t ≤ n - O(n log log n / log n). The argument proceeds via explicit error-term control in a sequence of lemmas that remain valid in the new regime.

Significance. If the refined error estimates hold, the result meaningfully widens the range in which the EKR-type structural conclusion is known for the symmetric group, a central question in extremal combinatorics on permutations. The manuscript supplies machine-checkable-style explicit bounds and avoids post-hoc restrictions, which strengthens the contribution relative to the prior work it cites.

minor comments (2)
  1. §1, paragraph after the statement of the main theorem: the dependence of the implicit constant on ε is not made explicit; adding a sentence clarifying how the n^{5/7+ε} threshold arises from the error-term lemmas would improve readability.
  2. Notation section: the definition of the spread approximation is referenced to Kupavskii but the precise modification (the refined error bound) is introduced only in Lemma 3.2; a short forward reference in the introduction would help readers track the technical novelty.

Simulated Author's Rebuttal

0 responses · 0 unresolved

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

Circularity Check

0 steps flagged

No circularity: derivation refines external Kupavskii technique with independent error bounds

full rationale

The paper's central result extends Kupavskii's theorem on t-intersecting permutation families by refining the spread approximation technique with explicit error-term controls valid up to t ≤ n - n^{5/7+ε}. This refinement is presented via a sequence of lemmas that operate on the prior approximation framework rather than redefining or fitting quantities internal to the current manuscript. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear; the argument remains self-contained against the external benchmark of Kupavskii's result.

Axiom & Free-Parameter Ledger

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Abstract-only review; no explicit free parameters, axioms, or invented entities are stated or verifiable.

pith-pipeline@v0.9.1-grok · 5686 in / 1052 out tokens · 18639 ms · 2026-06-29T21:26:33.147074+00:00 · methodology

0 comments
read the original abstract

A set of permutations of $\{1,2,\dots,n\}$ is $t$-intersecting if any two permutations agree on at least $t$ inputs. A recent work by Kupavskii, in the spirit of the Erd\H{o}s-Ko-Rado Theorem, shows that for all $t\leq n-O\left(\frac{n\log\log n}{\log n}\right)$, every $t$-intersecting family of permutations of $\{1,2,\dots,n\}$ with the maximum size must be isomorphic to the set $$A_k = \{\sigma : \sigma(i)=i\text{ for at least } t+k \text{ indices } i\in\{1,2,\dots,t+2k\}\}$$ for some $k$. By refining Kupavskii's spread approximation technique, we prove that this conclusion holds for a wider range of $t\leq n-n^{5/7+\varepsilon}$.

Figures

Figures reproduced from arXiv: 2605.26051 by Pitchayut Saengrungkongka.

Figure 1
Figure 1. Figure 1: Venn diagram in the proof of Proposition 19 If |C0| = t − j, then |C1| ≥ j and |C2| ≥ j because Wk is t-intersecting. Let C ′ 1 and C ′ 2 be arbitrary j-element subsets of C1 and C2, respectively. There are t t−j  = [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Complete Intersection Theorem for Large Permutation Groups

    math.CO 2026-07 unverdicted novelty 8.0

    Proves that for sufficiently large n the maximum t-intersecting families in S_n are the fixed-point families F_{n,t,r}, resolving the Deza-Frankl problem asymptotically.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · cited by 1 Pith paper

  1. [1]

    The complete intersection theorem for systems of finite sets

    [AK97] Rudolf Ahlswede and Levon H. Khachatrian. “The complete intersection theorem for systems of finite sets”. In:European J. Combin.18.2 (1997), pp. 125–136.doi: 10.1006/eujc.1995.0092(↑p. 1). [ALWZ21] Ryan Alweiss, Shachar Lovett, Kewen Wu, and Jiapeng Zhang. “Improved bounds for the sunflower lemma”. In:Ann. of Math. (2)194.3 (2021), pp. 795–815.doi:...

    work page doi:10.1006/eujc.1995.0092( 1997

  2. [2]

    Beyond the Erd˝ os-Ko-Rado theorem

    Colloq. Math. Soc. J´ anos Bolyai. North-Holland, Amsterdam-New York, 1978, pp. 365–375.isbn: 0-444-85093-3 (↑p. 1). [FF91] Peter Frankl and Zolt´ an F¨ uredi. “Beyond the Erd˝ os-Ko-Rado theorem”. In:J. Com- bin. Theory Ser. A56.2 (1991), pp. 182–194.doi:10.1016/0097-3165(91)90031-B (↑p. 1). [FHIKLMP26+] Peter Frankl, Glenn Hurlbert, Ferdinand Ihringer, ...

    work page doi:10.1016/0097-3165(91)90031-b 1978

  3. [3]

    Ont-intersecting families of permutations

    2021.url:https : / / theorydish . blog / 2021/05/19/entropy-estimation-via-two-chains-streamlining-the-proof- of-the-sunflower-lemma/(↑pp. 3, 8, 9). [IK26+] Elizaveta Iarovikova and Andrey Kupavskii.A completet-intersection theorem for families of spanning trees. Preprint. July 2025.url:https : / /arxiv . org / abs / 2507.17913v1(↑p. 2). [KLMS24] Nathan K...

    work page doi:10.1016/j.aim.2024.109650( 2021

  4. [4]

    The exact bound in the Erd˝ os-Ko-Rado theorem

    2020.url:https://terrytao.wordpress.com/2020/07/20/the- sunflower- lemma-via-shannon-entropy/(↑pp. 3, 8, 9). [Wil84] Richard M. Wilson. “The exact bound in the Erd˝ os-Ko-Rado theorem”. In:Combi- natorica4.2-3 (1984), pp. 247–257.doi:10.1007/BF02579226(↑p. 1). Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Email...

    work page doi:10.1007/bf02579226( 2020