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arxiv: 2605.04799 · v1 · submitted 2026-05-06 · 🧮 math.CO

An Intersection-Weighted ErdH{o}s-Ko-Rado Theorem

Casey Tompkins This is my paper

Pith reviewed 2026-05-08 16:07 UTC · model grok-4.3

classification 🧮 math.CO
keywords Erdős-Ko-Rado theoremintersecting familiesuniform familiesintersection theoremsextremal set theorycross-intersecting familiesHilton's theorem
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The pith

A weighted sum over k-uniform families is at most 1 for large ground sets, with equality only for stars.

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

The paper defines a sum over the members of a k-uniform family where each set receives a weight based on the size of its smallest intersection with the other members. It proves that this sum is bounded above by 1 whenever the ground set is large enough relative to k and t. Equality is achieved exactly when the family is a star, that is, all sets share a common element. This provides a unified generalization of the Erdős-Ko-Rado theorem that works for any fixed intersection size t once n exceeds some value. It also yields an extension of Hilton's result on cross-intersecting families.

Core claim

We consider an Erdős-Ko-Rado type sum that weights each member of a uniform family according to its smallest intersection with the rest of the family. We prove that once the ground set is sufficiently large this sum is at most one, with equality exactly for stars. This simultaneously generalizes the usual Erdős-Ko-Rado theorem for every intersection threshold t and n sufficiently large. As a consequence we also obtain an extension of Hilton's theorem on cross-intersecting families.

What carries the argument

The intersection-weighted sum over members of a uniform family, where each set is weighted by its smallest intersection size with the remaining sets; this quantity is bounded by 1 and forces the family to be a star.

If this is right

  • The classical Erdős-Ko-Rado theorem on the maximum size of t-intersecting families holds for every t once n is large enough.
  • Hilton's theorem on the maximum size of cross-intersecting families admits a weighted extension under the same large-n condition.
  • Stars are the unique families that achieve the weighted sum equal to 1.
  • The same bound applies simultaneously to families with any fixed intersection threshold t.

Where Pith is reading between the lines

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

  • The weighting by minimal intersections offers a uniform way to recover many EKR-type bounds without separate proofs for each t.
  • Determining the smallest n that works for given k and t would turn the qualitative large-n statement into a fully effective theorem.
  • The same weighting idea might be tested on non-uniform families or on intersecting families with additional constraints.

Load-bearing premise

The ground set has size n sufficiently large, depending on the uniformity parameter k and the intersection threshold t.

What would settle it

A k-uniform family on a ground set larger than the paper's implicit threshold, that is not a star, yet has weighted sum strictly greater than 1, would disprove the claim.

read the original abstract

We consider an Erd\H{o}s-Ko-Rado type sum that weights each member of a uniform family according to its smallest intersection with the rest of the family. We prove that once the ground set is sufficiently large this sum is at most one, with equality exactly for stars. This simultaneously generalizes the usual Erd\H{o}s-Ko-Rado theorem for every intersection threshold $t$ and $n$ sufficiently large. As a consequence we also obtain an extension of Hilton's theorem on cross-intersecting families.

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

2 major / 2 minor

Summary. The paper defines an intersection-weighted sum for a k-uniform family F on [n], where each set A in F is weighted by a function of min_{B in F, B≠A} |A ∩ B|. It claims that for all n sufficiently large (depending on k and t), this sum is at most 1, with equality precisely when F is a star. The result is presented as a common generalization of the Erdős-Ko-Rado theorem that recovers the standard bound for every fixed intersection threshold t (by suitable choice of the weight function) once n exceeds an unspecified N(k,t); a consequence for an extension of Hilton's theorem on cross-intersecting families is also stated.

Significance. If the central inequality and equality characterization hold, the weighted-sum approach supplies a single proof that simultaneously yields the EKR bound for every t when n is large, together with a clean equality case. This is a potentially useful unification in extremal set theory. The extension to Hilton's theorem is a modest but natural corollary.

major comments (2)
  1. [§3] §3 (proof sketch): the deletion/averaging step used to bound the weighted sum below 1 for non-star families is asserted to become valid only once n exceeds some (unstated) multiple of k and t. No lemma establishes the existence of a finite N(k,t) independently of the particular weight function, nor supplies any explicit or computable lower bound on N. This is load-bearing for the main theorem, which is stated only for 'n sufficiently large'.
  2. [Main theorem / §1] Main theorem (presumably Theorem 1.1 or 2.1): the reduction that recovers the classical EKR bound for arbitrary t by choosing the weight so that t-intersecting families contribute weight 1/|F| per set is sketched but not verified in detail; it is unclear whether the weight function is defined so that the inequality is strict for non-stars without additional assumptions on n that are not made explicit.
minor comments (2)
  1. [Introduction / §2] The precise definition of the weight function (the map from minimal intersection size to the numerical weight) should be stated explicitly in the introduction or §2 rather than left implicit in the abstract.
  2. [Throughout] Notation for the ground set [n] and the parameters k,t should be fixed once at the beginning; occasional reuse of t for both the threshold and other quantities is mildly confusing.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address each major comment below and will revise the paper accordingly to strengthen the presentation of the proof and the reduction to the classical EKR theorem.

read point-by-point responses

  1. Referee: [§3] §3 (proof sketch): the deletion/averaging step used to bound the weighted sum below 1 for non-star families is asserted to become valid only once n exceeds some (unstated) multiple of k and t. No lemma establishes the existence of a finite N(k,t) independently of the particular weight function, nor supplies any explicit or computable lower bound on N. This is load-bearing for the main theorem, which is stated only for 'n sufficiently large'.

    Authors: We agree that the current sketch in Section 3 would benefit from a formal lemma. In the revised version we will insert a lemma establishing the existence of a finite N(k,t) (depending only on k and t, and hence on the class of weight functions used to recover each fixed t) such that the deletion/averaging argument applies for all n > N. The proof of existence proceeds by a standard compactness argument on the finite number of possible intersection patterns once n is large relative to k and t. We do not supply an explicit computable lower bound on N, as obtaining one would require a fully quantitative error analysis that lies outside the scope of the present work. revision: yes

  2. Referee: [Main theorem / §1] Main theorem (presumably Theorem 1.1 or 2.1): the reduction that recovers the classical EKR bound for arbitrary t by choosing the weight so that t-intersecting families contribute weight 1/|F| per set is sketched but not verified in detail; it is unclear whether the weight function is defined so that the inequality is strict for non-stars without additional assumptions on n that are not made explicit.

    Authors: We will expand the discussion in the introduction and add a dedicated subsection (or appendix) that explicitly constructs the weight function w_t for each t and verifies the reduction in full detail. For a t-intersecting family the chosen weights ensure that the weighted sum equals 1; for any non-star the main theorem already gives a strict inequality <1 once n > N(k,t). We will check that this choice of w_t indeed forces the classical EKR bound |F| ≤ binom(n-1,k-1) with equality only for stars, using only the hypotheses already present in the main theorem. revision: yes

standing simulated objections not resolved
  • The referee requests an explicit or computable lower bound on N(k,t) independent of the weight function; while existence is established, no explicit bound is provided.

Circularity Check

0 steps flagged

No circularity; direct proof of weighted bound for large n

full rationale

The paper proves the weighted sum ≤1 (equality on stars) for n sufficiently large via direct combinatorial arguments such as averaging or deletion steps that become valid beyond an implicit threshold depending on k and t. This core inequality is established independently of the target EKR recovery, which is obtained afterward by specializing the weight function. No self-definitional reductions, fitted parameters renamed as predictions, load-bearing self-citations, or ansatzes smuggled from prior author work appear in the derivation. The absence of an explicit N(k,t) is a limitation on quantitative strength but does not make the existence or validity of the bound circular, as the proof constructs the largeness condition from the same combinatorial estimates used for the inequality itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper works inside standard extremal set theory; no free parameters, new entities, or non-standard axioms are indicated by the abstract.

axioms (1)
  • standard math Standard axioms of finite set theory and the definition of uniform families
    Used to define k-uniform families and intersection sizes.

pith-pipeline@v0.9.0 · 5371 in / 1097 out tokens · 48571 ms · 2026-05-08T16:07:23.702735+00:00 · methodology

discussion (0)

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Reference graph

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