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 $r\ge 2$ 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 $k\ge 1$, 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. As a consequence, we obtain a version of the classical Erdős--Ko--Rado theorem in the $(r-1,k)$-norm.

16 pages