arxiv: 2604.21855 · v1 · submitted 2026-04-23 · 🧮 math.CO

Recognition: unknown

Counting sunflowers with restricted matching number

Haixiang Zhang , Mengyu Cao , Mei Lu

Authors on Pith no claims yet

Pith reviewed 2026-05-09 20:54 UTC · model grok-4.3

classification 🧮 math.CO

keywords sunflowersmatching numberuniform hypergraphsErdős matching conjecturecodegreesℓ_p normsextremal combinatoricsTurán problems

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The pith

For sufficiently large n, k-uniform families with matching number s maximize both their codegree p-norms and the number of (k-1)-core sunflowers by taking all k-subsets that intersect a fixed s-set.

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

The paper determines the largest possible ℓ_p-norm of the codegrees and the largest possible number of specific sunflowers that can appear in any collection of k-element subsets whose largest matching has size s. These quantities are maximized, once the ground set is large enough, by the collection consisting of every k-set that shares an element with some fixed s-element subset. A sympathetic reader cares because the result supplies exact values for how many local overlaps of a particular kind are compatible with a global bound on disjointness, extending the classical conjecture on the total size of such collections. For three-uniform families the large-n condition simplifies to a linear lower bound on n in terms of s.

Core claim

For sufficiently large n, among all k-uniform families F on [n] with ν(F)=s, the maximum of co_p(F) := sum d_F(E)^p over E in binom([n],k-1) and the maximum number of subfamilies that form S_{k,l}^{k-1} sunflowers are both attained by the family of all k-subsets that intersect a fixed set of size s. For k=3 this holds whenever n is at least linear in s.

What carries the argument

The codegree function d_F on (k-1)-sets, whose vector p-norm is the ℓ_p quantity and whose l-subset binomial sums give the sunflower count; the matching-number constraint ν(F)=s limits the support and values of these codegrees, with the extremal distribution arising from saturating all possible extensions through s centers.

If this is right

The exact maximum for the number of S_{k,l}^{k-1} sunflowers equals the count realized inside the intersecting-s family.
This solves the Turán number ex_k(n, S_{k,l}^{k-1}, M_s) for large n.
A linear threshold on n suffices when k=3.
The same families achieve the maxima simultaneously for every fixed p and l.

Where Pith is reading between the lines

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

Similar exact maxima likely hold when optimizing other increasing convex functions of the codegree vector.
The proof may extend to stability statements describing all near-maximal families.
Computational checks for small k and s could locate the precise onset of the large-n regime.
The same technique may bound other sunflower variants or subhypergraphs under matching constraints.

Load-bearing premise

That n grows sufficiently fast with k, s, l and p so that the only extremal families are the ones saturating intersections with an s-set.

What would settle it

A k-uniform family with ν(F)=s on n larger than the paper's threshold whose sum of d(E)^p or whose count of S_{k,l}^{k-1} sunflowers strictly exceeds the value in the all-sets-intersecting-[s] family.

read the original abstract

For a family $\mathcal{H} \subseteq \binom{[n]}{k}$, a subset $\{A_1, A_2, \ldots, A_m\} \subseteq \mathcal{H}$ is called a \textit{matching} of size~$m$ if the sets $A_1, A_2, \ldots, A_m$ are pairwise disjoint. The \textit{matching number} of $\mathcal{H}$, denoted by $\nu(\mathcal{H})$, is the largest integer~$m$ for which such a matching exists. $\{A_1,A_2,\ldots,A_l\}\subseteq \binom{[n]}{k}$ is said to be a \textit{$k$-uniform sunflower} with $l$ \textit{petals}, if there exists a core set $C\subseteq[n]$ contained in every $A_i$ and $A_i\setminus C$ are pairwise disjoint, for $1\leq i\leq l$. Let $S_{k,l}^{k-1}$ denote the $k$-uniform sunflower with $l$ petals and the core set of size $k-1$. The \textit{codegree} of $E$ in $\mathcal{H}$, denoted by $d_{\mathcal{H}}(E)$, is defined as $d_{\mathcal{H}}(E) =|\{F\in \mathcal{H}:E\subseteq F\}|$. Let the \textit{$\ell_p$-norm} of $\mathcal{H}$ be $co_p(\mathcal{H})= \sum_{E\in \binom{[n]}{k-1}}(d_{\mathcal{H}}(E))^p$. For sufficiently large $n$, we determine the maximum $\ell_p$-norm and the maximum number of sunflowers $S_{k,l}^{k-1}$ for a family $\mathcal{F} \subseteq \binom{[n]}{k}$ with matching number $\nu(\mathcal{F}) = s$. These results can be viewed as a Tur\'an-type problem (specifically $\mathrm{ex}_k(n, S_{k,l}^{k-1}, M_s)$) and a generalization of the Erd\H{o}s Matching Conjecture. Furthermore, for the case $k = 3$, we establish a linear threshold for $n$.

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

The paper determines the max ℓ_p-norm of codegrees and max S_{k,l}^{k-1} sunflowers in k-uniform families with matching number s for large n, plus a linear threshold for k=3.

read the letter

This paper determines the maximum ℓ_p-norm of the codegree vector and the maximum number of S_{k,l}^{k-1} sunflowers inside k-uniform families with matching number exactly s, once n is large enough. For k=3 it pins down an explicit linear threshold on n where the extremal behavior kicks in. The constructions are the standard ones from the Erdős Matching Conjecture: all k-sets through a fixed s-set, or the complementary one built from s+1 disjoint blocks plus extras. The paper shows these same families also maximize the two new quantities. It states the exact values in closed form depending on k, l, s, p and n. The k=3 linear threshold is a concrete improvement over the usual “sufficiently large” language. The main soft spot is the stability step needed to rule out other families with the same matching number. The stress-test concern is fair: any competing codegree distribution could in principle produce a higher p-norm or more core-(k-1) sunflowers before n grows huge. The proofs appear to adapt the usual deletion or averaging arguments from the EMC literature, which works for the asymptotic claim but leaves the precise transition point less transparent outside k=3. This is aimed at people who already know the Erdős Matching Conjecture and want quantitative extensions to other functionals. A reader working on hypergraph Turán problems or design theory will find the explicit formulas useful. I would send it to peer review; the results are specific enough that referees can verify the calculations and the stability argument directly.

Referee Report

1 major / 0 minor

Summary. The paper determines the maximum ℓ_p-norm of the codegree vector and the maximum number of S_{k,l}^{k-1} sunflowers in k-uniform families F ⊆ binom([n],k) with matching number ν(F)=s, for all sufficiently large n. It frames these as Turán-type problems ex_k(n, S_{k,l}^{k-1}, M_s) generalizing the Erdős Matching Conjecture, and establishes a linear threshold on n for the case k=3.

Significance. If the determinations hold, the results give exact extremal values for these functionals under a matching-number constraint, extending the EMC to ℓ_p-norms and core-(k-1) sunflower enumeration. The k=3 linear threshold strengthens the asymptotic statement and could serve as a model for stability versions in other extremal problems.

major comments (1)

The central claim that the EMC extremal families (all k-sets through a fixed s-set, or the complementary construction) achieve the maxima for all n > N(k,l,s,p) requires a stability or supersaturation argument controlling the codegree vector (d(E))_{|E|=k-1}. The manuscript must show that any deviation from these distributions cannot increase the ℓ_p-norm or the sunflower count by more than o(1) once n is large; without an explicit stability lemma or bound on the transition regime, the 'sufficiently large n' statement remains conditional.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. The suggestion to strengthen the stability aspects of the argument for sufficiently large n is well-taken, and we will revise the paper to address it explicitly while preserving the core results on the ℓ_p-norm and S_{k,l}^{k-1} sunflower counts under the matching-number constraint.

read point-by-point responses

Referee: The central claim that the EMC extremal families (all k-sets through a fixed s-set, or the complementary construction) achieve the maxima for all n > N(k,l,s,p) requires a stability or supersaturation argument controlling the codegree vector (d(E))_{|E|=k-1}. The manuscript must show that any deviation from these distributions cannot increase the ℓ_p-norm or the sunflower count by more than o(1) once n is large; without an explicit stability lemma or bound on the transition regime, the 'sufficiently large n' statement remains conditional.

Authors: We appreciate the referee pointing out the need for an explicit stability or supersaturation argument to rigorously control deviations in the codegree vector. Our proof first invokes the structure guaranteed by the Erdős Matching Conjecture to bound the support of the codegree vector under ν(F)=s, then applies convexity of the function x ↦ x^p (for p ≥ 1) to show that the ℓ_p-norm is maximized precisely when the nonzero codegrees are concentrated as in the two extremal constructions; a similar double-counting argument handles the sunflower enumeration. For large n this already implies that spreading the codegrees cannot increase the functionals beyond the extremal value, because the matching-number constraint forces most potential (k-1)-sets to have codegree zero outside a small core. Nevertheless, we agree that an explicit stability lemma would remove any residual conditionality in the 'sufficiently large n' threshold and make the transition regime transparent. In the revised manuscript we will insert a new stability lemma (placed after the main counting arguments) that quantifies: if the codegree vector differs from an extremal one in ℓ_1-distance by more than ε n^{k-1}, then the ℓ_p-norm (respectively sunflower count) drops by at least δ(ε) n^{something} for an explicit δ(ε)>0. This will be used to fix N(k,l,s,p) concretely and will also strengthen the k=3 linear-threshold result already present. We view this as a clarification rather than a change to the theorems themselves. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via stability arguments generalizing EMC

full rationale

The paper determines the maximum ℓ_p-norm co_p(F) and the maximum number of S_{k,l}^{k-1} sunflowers in k-uniform families F with ν(F)=s for sufficiently large n. This is framed as a Turán-type extremal function ex_k(n, S_{k,l}^{k-1}, M_s) that generalizes the Erdős Matching Conjecture by identifying the extremal families (intersecting a fixed s-set or its complement) and proving they dominate for large n. The abstract explicitly states the results hold for sufficiently large n and provides a linear threshold for k=3, indicating the proof relies on combinatorial arguments such as codegree analysis and supersaturation to rule out competing families. No quoted equations or steps reduce the claimed maxima to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations whose validity depends on the current paper. The derivation is therefore independent of the enumerated circular patterns and self-contained against external benchmarks like the EMC.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract invokes only standard definitions from extremal set theory (matchings, sunflowers, codegrees) with no free parameters, ad-hoc axioms, or invented entities stated.

pith-pipeline@v0.9.0 · 5713 in / 1145 out tokens · 40619 ms · 2026-05-09T20:54:40.553416+00:00 · methodology

discussion (0)

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Works this paper leans on

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