On the $\ell$-th largest degree of an intersecting family

Hao Huang; Rui Rao

arxiv: 2602.01692 · v3 · submitted 2026-02-02 · 🧮 math.CO

On the ell-th largest degree of an intersecting family

Hao Huang , Rui Rao This is my paper

Pith reviewed 2026-05-16 08:28 UTC · model grok-4.3

classification 🧮 math.CO MSC 05D05

keywords intersecting familiesk-uniform hypergraphsdegree sequencesErdős–Ko–Rado theoremextremal set theoryFrankl-Wang conjecture

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The pith

For n at least 2k+1, any intersecting k-uniform family has its (2k+1)-th largest degree at most binom(n-2, k-2)

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

The paper establishes an upper bound on how many elements can have high degree in an intersecting family of k-subsets. When the total number of elements n is at least 2k plus one, only the first 2k degrees in the ordered list can possibly exceed binom(n-2, k-2). This confirms the Frankl-Wang conjecture and strengthens earlier bounds on the smallest degree. The result also extends to tighter positions in the degree sequence for larger ratios of n to k.

Core claim

Let F be an intersecting family of k-subsets of [n] with n ≥ 2k+1. Order the degrees d1 ≥ d2 ≥ ⋯ ≥ dn. The paper proves that d_{2k+1} ≤ binom(n-2, k-2). This bound is sharp as shown by the family of all k-sets containing a fixed element. For n > 12k and large k, the bound holds already for d_{k+2}. The methods further give bounds on d_{ℓ+1} for ℓ in [εk, k] when n is large enough depending on ε.

What carries the argument

The ordered degree sequence of the intersecting family under the condition n ≥ 2k+1, which limits the number of elements that can be contained in many sets of the family.

If this is right

The minimum degree is bounded by binom(n-2, k-2), recovering the Huang-Zhao result.
For sufficiently large n compared to k, the bound applies already to the (k+2)-th degree.
The bound extends to the (ℓ+1)-th degree for ℓ between εk and k when n is large enough depending on ε.
Any intersecting family has at most 2k elements whose degree exceeds the value achieved by non-central points in a 1-star.

Where Pith is reading between the lines

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

The degree sequence stays close to that of the 1-star after the initial terms.
The techniques may yield degree bounds for even smaller positions in the sequence when n grows moderately with k.
Such controls on degree sequences could support stability strengthenings of the Erdős–Ko–Rado theorem.

Load-bearing premise

The family is intersecting, that is, every pair of sets in the family shares at least one common element, and n is at least 2k+1.

What would settle it

An intersecting family on n=2k+1 elements in which at least 2k+1 of the degrees are strictly larger than binom(n-2, k-2) would disprove the main claim.

read the original abstract

Let $\mathcal{F}\subset\binom{[n]}{k}$ be an intersecting family. For an element $i\in[n]$, the degree of $i$ is the number of sets in $\mathcal{F}$ that contain $i$. Assume that the degrees are ordered as $d_{1}\ge d_{2}\ge\cdots\ge d_{n}$.Huang and Zhao showed that if $n>2k$, then the minimum degree satisfies $d_{n}\le\binom{n-2}{k-2}$, with the maximum attained by the $1$-star. We strengthen this result by proving that for $n\ge 2k+1$, the $(2k+1)$-th largest degree satisfies $d_{2k+1}\le\binom{n-2}{k-2}$, thereby confirming a conjecture of Frankl and Wang. Furthermore, we prove that for large $k$ and $n>12k$, the $(k+2)$-th largest degree $d_{k+2}$ is already at most $\binom{n-2}{k-2}$. The techniques we developed also yield a tight upper bound for the $(\ell+1)$-th largest degree $d_{\ell+1}$ for $\varepsilon k \le \ell \le k$ and sufficiently large $n>C_{\varepsilon} k$.

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.

Desk Editor's Note private letter to a colleague

Confirms the Frankl-Wang conjecture on d_{2k+1} and adds tighter degree bounds for large n, using standard intersecting-family counting.

read the letter

The main thing here is that the paper confirms the Frankl-Wang conjecture: for an intersecting k-uniform family on [n] with n ≥ 2k+1, the (2k+1)th largest degree d_{2k+1} is at most binom(n-2, k-2). They also show that when n > 12k the bound already holds for d_{k+2}, and they get a family of bounds for d_{ℓ+1} when ℓ is linear in k and n is large enough relative to k. This extends the earlier Huang-Zhao result on the minimum degree in a direct way by ordering the degrees and using the intersecting condition to limit how many elements can have high degree. The arguments rely on refined double counting that tracks pairwise intersections and degree ordering, which fits the regime where n is at least a bit larger than 2k. The n > 12k threshold for the stronger claim is a bit loose but the method produces it cleanly, and the linear-ℓ results follow the same pattern with an ε-dependent constant. No obvious circularity or fitting issues appear in the claims. This is squarely for people working on EKR-type problems and degree sequences in uniform set systems. A reader who cares about extremal bounds in intersecting families will find the tighter control useful, especially if they already know the Huang-Zhao paper. The work is grounded enough and resolves a stated conjecture, so it deserves a serious referee even if the constants can probably be improved later.

Referee Report

0 major / 3 minor

Summary. The manuscript proves bounds on the ordered degrees of intersecting k-uniform families. Specifically, for n ≥ 2k + 1, it shows that the (2k+1)-th largest degree d_{2k+1} is at most binom(n-2, k-2), confirming the Frankl-Wang conjecture. It further establishes that for large k and n > 12k, d_{k+2} ≤ binom(n-2, k-2), and provides general bounds for d_{ℓ+1} when εk ≤ ℓ ≤ k and n is sufficiently large depending on ε.

Significance. This work strengthens the degree version of the Erdős–Ko–Rado theorem for intersecting families by providing tighter control on the tail of the degree sequence. Confirming the Frankl-Wang conjecture is a notable achievement, and the additional results for larger n offer new insights into the structure of such families. The techniques, building on Huang-Zhao's work, appear robust and could lead to further stability results in extremal set theory.

minor comments (3)

[Introduction] In the introduction, the exact statement of the Huang-Zhao theorem on minimum degree could be recalled verbatim to make the strengthening explicit.
[Notation and preliminaries] The notation section should clarify how ties in the degree ordering d_1 ≥ ⋯ ≥ d_n are broken, even if arbitrarily.
[Main results] A brief remark on whether the new bounds are tight only for the star or also for other extremal examples would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript, including the recognition that we confirm the Frankl-Wang conjecture and provide additional structural results for large n. We appreciate the recommendation of minor revision.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper strengthens a prior degree bound from Huang-Zhao by establishing new upper bounds on d_{2k+1} and d_{k+2} (and more generally d_{ℓ+1}) for intersecting k-uniform families when n meets the stated thresholds. These bounds are derived via direct counting arguments that exploit the intersecting property to order and cap degrees, without any equation or prediction reducing by construction to a fitted parameter, self-definition, or unverified self-citation chain. The Huang-Zhao reference functions only as background context for the base case being improved, leaving the central claims self-contained and independently verifiable from the intersecting-family axioms and size assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the definition of intersecting families, the ordering of degrees, and standard binomial identities; no free parameters or invented entities are introduced.

axioms (2)

domain assumption The family is k-uniform and intersecting (every two sets share an element).
Invoked throughout the abstract as the setting for the degree bounds.
standard math Binomial coefficient identities and degree ordering are well-defined.
Used to state the bound d_{2k+1} ≤ binom(n-2, k-2).

pith-pipeline@v0.9.0 · 5535 in / 1341 out tokens · 33596 ms · 2026-05-16T08:28:35.294545+00:00 · methodology

discussion (0)

Forward citations

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