论文标题
具有指数为许多极端结构的非分级超图
Non-degenerate Hypergraphs with Exponentially Many Extremal Constructions
论文作者
论文摘要
对于每个整数$ t \ ge 0 $，用$ f_5^t $表示顶点套装$ \ {1,2，\ ldots，5+t \} $，带有hyperedges $ \ {123,124 \ \ \ \ \ \ \ \ \ \ \ \ \} \ cup \ cup \ cup \ {34K：5 \ le K \ le k \ le k \ le k \ le 5+t \ t \ t \} $。我们确定$ \ mathrm {ex}（n，f_5^t）$的每个$ t \ ge 0 $和足够大的$ n $，并且表征了极端$ f_5^t $ -free HyperGraphs。特别是，如果$ n $满足某些划分条件，则极端$ f_5^t $ - 免费的超图完全是平衡完整的三方超图，并在分区中的三个部分$（v_1，v_2，v_3）$中的每个零件内部有其他超级货币；每个部分$ v_i $跨越$（| v_i |，3,2，t）$ - 设计。这概括了弗兰克尔（Frankl）和füredi在$ f_5的turán数字上的早期工作：= f_5^0 $。 我们的结果扩展了有关某些固定图的极端结构的ERD和Simonovits的理论。特别是，对于$ t \ geq 1 $，超图$ f_5^{6t} $是具有指数级的许多极端结构和正turán密度的超图的示例。
For every integer $t \ge 0$, denote by $F_5^t$ the hypergraph on vertex set $\{1,2,\ldots, 5+t\}$ with hyperedges $\{123,124\} \cup \{34k : 5 \le k \le 5+t\}$. We determine $\mathrm{ex}(n,F_5^t)$ for every $t\ge 0$ and sufficiently large $n$ and characterize the extremal $F_5^t$-free hypergraphs. In particular, if $n$ satisfies certain divisibility conditions, then the extremal $F_5^t$-free hypergraphs are exactly the balanced complete tripartite hypergraphs with additional hyperedges inside each of the three parts $(V_1,V_2,V_3)$ in the partition; each part $V_i$ spans a $(|V_i|,3,2,t)$-design. This generalizes earlier work of Frankl and Füredi on the Turán number of $F_5:=F_5^0$. Our results extend a theory of Erdős and Simonovits about the extremal constructions for certain fixed graphs. In particular, the hypergraphs $F_5^{6t}$, for $t\geq 1$, are the first examples of hypergraphs with exponentially many extremal constructions and positive Turán density.
