Counting sunflowers in hypergraphs with bounded matching number and Erdős Matching Conjecture in the $(t,k)$-norm

Abstract: It is well known that Erdős Matching Conjecture concerns the maximum number of hyperedges in an $r$-uniform hypergraph with bounded matching number. As a generalization, it is natural to ask for the maximum number of copies of subhypergraphs. Given integers $r\geq2$ and $k\ge 1$, let $S_{r-1,k}^r$ denote the $r$-uniform hypergraph with hyperedges ${e_1, \dots, e_k}$ such that there exists an $(r-1)$-set $T$ with $e_i \cap e_j = T$ for $1\le i < j \le k$. We determine the maximum number of copies of $S_{r-1,k}^r$ in an $r$-uniform hypergraph with bounded matching number, and characterize all extremal hypergraphs. An interesting phenomenon is that the extremal numbers and extremal hypergraphs are exactly the same for all $k\ge 1$. Our main tool is the shifting method. By establishing an injection, we prove that the shifting operation does not decrease the number of copies of $S_{r-1,k}^r$ for all $k\geq1$, thereby answering a question raised by Wang and Peng (2026). Moreover, we present a counting method for estimating the number of copies of $S_{r-1,k}^r$ in arbitrary $r$-uniform hypergraphs. Counting the number of copies of $S_{r-1,k}^r$ in $r$-uniform hypergraphs is closely related to Turán problems in the $(r-1,k)$-norm proposed by Chen, Il'kovič, León, Liu and Pikhurko. The $(r-1,k)$-norm of an $r$-uniform hypergraph $\mathcal{H}$ is the sum of the $k$-th power of the degrees $d_{\mathcal{H}}(T)$ over all $(r-1)$-subsets $T \subseteq V(\mathcal{H})$. Combining our established result with that of Frankl (2013), and utilizing the Newton expansion of powers and Stirling numbers of the second kind, we show that Erdős Matching Conjecture in the $(r-1,k)$-norm holds, which generalizes the result of Brooks and Linz concerning the $(r-1,2)$-norm case.

• The paper establishes the unique extremal hypergraph construction that maximizes sunflower counts in r-uniform hypergraphs under bounded matching constraints.
• It extends the shifting method to prove that sunflower counts for S₍r₋1,k₎ʳ do not decrease, ensuring robust structural analysis.
• The study generalizes the Erdős Matching Conjecture to the (r-1,k)-norm, providing explicit asymptotic bounds and deep insights into Turán-type problems.

Counting Sunflowers in Hypergraphs with Bounded Matching Number and the Erdős Matching Conjecture in the $(t, k)$-Norm

Introduction

This work presents a structural and quantitative study of sunflowers in $r$-uniform hypergraphs under matching constraints, leveraging the shifting method. The sunflower $S_{r-1,k}^r$ is defined as the $r$-graph formed by $k$ hyperedges that all share a common $(r-1)$-set (the core), and are otherwise disjoint (the petals). The main goal is to determine, for a fixed $r$, $k$, and matching bound $s$, the hypergraph(s) on $n$ vertices with the largest number of $r$0 copies. This generalizes previous results for the case $r$1.

The study is deeply connected not only to generalized Turán-type questions, which concern extremal substructure counts rather than edge counts, but also to an "average" variant—Turán problems under the $r$2-norm, introduced by Chen et al.—where the goal is to maximize a particular norm of the degree sequence over $r$3-sets, and whose combinatorial meaning is tied to counting sunflowers.

Main Results and Implications

Extremal Sunflower Counts in $r$4-Uniform Hypergraphs

Let $r$5 denote the matching of size $r$6 in $r$7-graphs, i.e., the hypergraph substructure formed by $r$8 pairwise disjoint $r$9-edges. The core theorem establishes that for all $S_{r-1,k}^r$0, the extremal number of $S_{r-1,k}^r$1 copies in $S_{r-1,k}^r$2-free $S_{r-1,k}^r$3-graphs on $S_{r-1,k}^r$4 vertices and sufficiently large $S_{r-1,k}^r$5 is achieved uniquely (up to isomorphism) by the initial segment hypergraph $S_{r-1,k}^r$6 (i.e., the family of all $S_{r-1,k}^r$7-sets meeting a fixed set of $S_{r-1,k}^r$8 vertices).

Strikingly, both the extremal numbers and the extremal constructions are invariant with respect to $S_{r-1,k}^r$9, generalizing and unifying the results for $r$0 and classical results about extremal edge numbers (the original Erdős Matching Conjecture).

These assertions are formalized in Theorem 1 and the Erdős Matching Conjecture in the $r$1-norm (Theorem 2).

The Shifting Method and Sunflower Monotonicity

A technical highlight is the extension of the shifting method to maximization of sunflower counts for all $r$2. The shifting operation is shown to never decrease the number of $r$3 copies (Lemma~\ref{newlem}), a fact that is not generally true for sunflowers with smaller cores or for other hypergraph families.

Figure 1: Shifting operations fail to preserve several expansion-freeness properties (e.g., for paths, cycles, stars, cliques), but for matchings they are safe.

Figure 2: For some general $r$4 (with $r$5), shifting can strictly decrease the sunflower count.

The paper constructs a detailed case analysis and injective mapping to verify the non-decreasing property for $r$6 under shifting, resolving a question raised in prior work by Wang and Peng.

Sunflower Counting and $r$7-Norm Optimization

Counting copies of $r$8 is tightly connected to optimization of the $r$9-norm

$k$0

which by Newton's identities can be decomposed as a weighted sum of the sunflower counts for all $k$1.

Thus, the paper confirms the Erdős Matching Conjecture for the $k$2-norm and describes all extremal hypergraphs for this norm (Theorem 2), generalizing previous work for $k$3 and classical ($k$4) cases.

Methodological Advances

The main technical tools are:

• Iterated Shifting: Shifting preserves $k$5-freeness (due to Frankl) and does not decrease $k$6 copy counts, justifying induction on the shifted class.
• Petal/Core Decomposition Counting: Detailed recurrences and decompositions for counting $k$7 copies in general hypergraphs (Lemma~3.1), sensitive to potential petal/core overlaps and the vertex of interest.
• Structural Reduction: Careful exploitation of the shifted property and edge-maximality to reduce to the unique extremal candidate set system.

Numerical Outcomes and Explicit Bounds

Strong asymptotic and sometimes explicit bounds are given for extremal counts and minimal required $k$8 (see, e.g., inequalities (3.2) and (3.3)). The case $k$9 can in fact give better lower bounds on $(r-1)$0 compared to the classical threshold.

Theoretical and Practical Implications

Theoretical: The results illuminate the landscape of generalized Turán-type extremal problems in hypergraphs, especially those invariant under shifting and reducible to highly symmetric initial segment hypergraphs. They also clarify for which subhypergraphs shifting is suitable (e.g., for $(r-1)$1 but generally not $(r-1)$2 with smaller cores).

Practical: Understanding the structure of extremal families for matching-bounded hypergraphs informs combinatorial optimization, design theory, and applications in randomized algorithms and information theory, where sunflowers play a role in derandomization and splitters.

Future Research: The authors raise open questions regarding the characterization of all families for which shifting preserves freeness or maintains subhypergraph counts, an area ripe for further exploration. Additionally, seeking explicit expressions for $(r-1)$3 and extending the approach to other norm-based extremal problems are prospective directions.

Conclusion

This work gives a comprehensive solution for maximizing sunflower counts in $(r-1)$4-graphs with bounded matching number, characterizes the unique extremal constructions, and resolves the Erdős Matching Conjecture under the $(r-1)$5-norm for all $(r-1)$6. The application of the shifting method is shown to be robust for these families, but its limitations are also carefully documented, opening several fruitful avenues for further theoretical development.