发表机构
Shanghai Jiao Tong University; The Hong Kong University of Science and Technology（上海交通大学; 香港科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究匹配数至多为\(s\)的\(r\) - 一致不含Berge - \(K_3\)超图的最大边数，确定\(r = 3\)和\(r = 4\)时的确切图兰数，刻画极值超图，还恢复了Győri关于不含Berge - \(K_3\)超图的经典定理。
AI 中文摘要
对于固定图\(G\)，若存在双射\(f\colon E(G)\to E(\mathcal{H})\)使得对每个\(e\in E(G)\)有\(e\subseteq f(e)\)，则称\(r\) - 一致超图包含一个Berge - \(G\)。受Alon和Frankl在有界匹配约束下对图兰问题研究的启发，我们研究匹配数至多为\(s\)的\(r\) - 一致不含Berge - \(K_3\)超图的最大边数。确定了\(r = 3\)和\(r = 4\)时的确切图兰数。对于\(r = 3\)且\(n\geq 3s\)，证明了每个匹配数为\(s\)的\(n\) - 顶点不含Berge - \(K_3\)的3 - 图至多有\(s(n - 2s)\)条边，并刻画了达到等式的唯一极值超图。对于\(r = 4\)且\(n\geq 4s\)，最大边数为\(s\lfloor(n - 2s)/2\rfloor\)，除了\(s = 1\)且\(n\equiv 1(\bmod 4)\)的特殊情况，此时边界为\((n - 1)/2\)。作为推论，我们的结果恢复了Győri关于不含Berge - \(K_3\)超图的经典定理。
英文摘要
For a fixed graph $G$, an $r$-uniform hypergraph is said to contain a Berge-$G$ if there exists a bijection $f\colon E(G)\to E(\mathcal{H})$ for some subhypergraph $\mathcal{H}$ such that $e\subseteq f(e)$ for every $e\in E(G)$. Motivated by Alon and Frankl's study of Turán problems under bounded matching constraints, we investigate the maximum number of edges in $r$-uniform Berge-$K_3$-free hypergraphs with matching number at most~$s$. We determine the exact Turán numbers for the cases $r=3$ and $r=4$. For $r=3$ and $n \geq 3 s$, we prove that every $n$-vertex Berge- $K_3$-free 3-graph with matching number $s$ has at most $s(n-2 s)$ edges, and we characterize the unique extremal hypergraph attaining equality. For $r=4$ and $n \geq 4 s$, the maximum number of edges is $s\lfloor(n-2 s) / 2\rfloor$, except for the exceptional case $s=1$ and $n \equiv 1(\bmod 4)$, in which the bound is $(n-1) / 2$. As a corollary, our results recover the classical theorem of Győri on Berge-$K_3$-free hypergraphs.
Comments23 pages
