Counting sunflowers in hypergraphs with bounded matching number and Erd\H{o}s Matching Conjecture in the $(t,k)$-norm

Abstract

It is well known that Erd\H{o}s 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 Tur\'{a}n problems in the $(r-1,k)$-norm proposed by Chen, Il'kovi\v{c}, Le\'{o}n, 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 Erd\H{o}s Matching Conjecture in the $(r-1,k)$-norm holds, which generalizes the result of Brooks and Linz concerning the $(r-1,2)$-norm case. As a consequence, we obtain a version of the classical Erd\H{o}s--Ko--Rado theorem in the $(r-1,k)$-norm.