论文标题
相交超图的双重多样性的最佳界限
Best possible bounds on the double-diversity of intersecting hypergraphs
论文作者
论文摘要
对于一个家庭，$ \ Mathcal {f} \ subset \ binom {[n]} {k} $和两个元素$ x，y \ in [n] $ define $ \ mathcal {f}（\ bar {x}}，\ bar {x}，\ bar {y}） y \ notin f \} $。双多样性$γ_2（\ MATHCAL {F}）$定义为$ | \ Mathcal {f}（\ bar {x}，\ bar {y}）| $上所有对$ x，y $。令$ \ Mathcal {l} \ subset \ binom {[7]} {3} $由Fano平面的七行组成。对于$ n \ geq 7 $，$ k \ geq 3 $一个人将fano $ k $ -graph $ \ mathcal {f} _ {\ mathcal {l}} $作为所有$ [n] $的所有$ k $ -subsets的集合。事实证明，对于$ n \ geq 13k^2 $，fano $ k $ -graph是本质上独特的家族，使所有$ k $ graphs均具有最大化的双重多样性，而没有一对脱节边缘。有些类似的结果也证明了三倍和更高多样性的结果。
For a family $\mathcal{F}\subset \binom{[n]}{k}$ and two elements $x,y\in [n]$ define $\mathcal{F}(\bar{x},\bar{y})=\{F\in \mathcal{F}\colon x\notin F,\ y\notin F\}$. The double-diversity $γ_2(\mathcal{F})$ is defined as the minimum of $|\mathcal{F}(\bar{x},\bar{y})|$ over all pairs $x,y$. Let $\mathcal{L}\subset\binom{[7]}{3}$ consist of the seven lines of the Fano plane. For $n\geq 7$, $k\geq 3$ one defines the Fano $k$-graph $\mathcal{F}_{\mathcal{L}}$ as the collection of all $k$-subsets of $[n]$ that contain at least one line. It is proven that for $n\geq 13k^2$ the Fano $k$-graph is the essentially unique family maximizing the double diversity over all $k$-graphs without a pair of disjoint edges. Some similar, although less exact results are proven for triple and higher diversity as well.
