A hypergraph analogue of Alon-Frankl Theorem
Caihong Yang, Jiasheng Zeng, Xiao-Dong Zhang
TL;DR
The work extends the Alon--Frankl paradigm to $r$-uniform hypergraphs by determining the exact Turán numbers for $\\mathcal{K}_{\\ell+1}^{r}$-free hypergraphs under a matching constraint, establishing that for large $n$ the maximum size is $s \\cdot t_{r-1}(n-s,\\ell-1)$ and is uniquely attained by the extremal construction $\\mathcal{G}(n,\\ell,s,r)$. The authors further solve the 3-uniform Fano plane case, giving an exact value and a unique extremal configuration, and discuss extensions to the $H_{\\ell+1}^{r}$ family with conjectures for broader regimes. The results rely on stability techniques, detailed analysis of link graphs, and a canonical partition of the vertex set into a size-$s$ core and a large weakly independent remainder, illuminating the structure of extremal hypergraphs under simultaneous forbiddance of a hypergraph family and a matching. This work advances hypergraph Turán theory by providing precise extremal configurations and enriching the understanding of how forbidding a cover-structure interacts with bounded matching.
Abstract
Recently, Alon and Frankl (JCTB, 2024) determined the maximum number of edges in $K_{\ell+1}$-free $n$-vertex graphs with bounded matching number. For integers $\ell\ge r \ge 2$, the family $\mathcal{K}_{\ell+1}^{r}$ consists of all $r$-graphs $F$ with at most $\binom{\ell+1}{2}$ edges such that, for some $(\ell+1)$-set $K$, every pair $\{x,y\} \subseteq K$ is covered by an edge in $F$. In this paper, we study the maximum number of edges in $\mathcal{K}_{\ell+1}^r$-free $r$-uniform hypergraphs that have the matching number at most $s$, that is, $\mathrm{ex}_r(n, \{\mathcal{K}_{\ell+1}^r, M^r_{s+1}\})$, and obtain the exact value for sufficiently large $n$, along with the corresponding extremal hypergraph. This result can be viewed as a hypergraph extension of the work of Alon and Frankl. In addition, for the $3$-uniform Fano plane $\mathbb{F}$, we determine the exact value of $\mathrm{ex}_3(n, \{\mathbb{F}, M^3_{s+1}\})$, and characterize the corresponding extremal hypergraph.