论文标题
相交超图中的破碎匹配
Shattered matchings in intersecting hypergraphs
论文作者
论文摘要
令$ x $为$ n $ element套件，其中$ n $均为。我们驳斥了J. Gordon和Y. Teplitskaya的猜想，根据这一点，对于每个最大相交的家庭$ \ Mathcal {f} $的$ \ frac {n} 2 $ element子集的$ x $的子集，每个人都可以将$ x $分配给$ \ frac {n} 2 $ nocation nody的extiage { $ \ frac {n} 2-1 $对，通过添加最后一对元素，可以始终将它们形成的集合完成给$ \ Mathcal {f} $的成员。 以上问题与极端集理论中的经典问题有关。对于任何$ t \ ge 2 $，我们称为$ \ nathcal {f} \ subset 2^x $ {\ em $ t $ -separable}的家族，如果有任何有序的元素$（x，y）$ x $的$（x，y）$ $ f \ cap \ {x，y \} = \ {x \} $。对于固定的$ t，2 \ le t \ le 5 $和$ n \ rightarrow \ infty $，我们对最小整数$ s = s（n，t）$建立了渐近的紧张估计，以使每个家庭$ \ mathcal {f} $均使用$ | \ m nathcal {f} f} | \ ge s $ $ t $ t $ t $ t $ -sseparoble。
Let $X$ be an $n$-element set, where $n$ is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family $\mathcal{F}$ of $\frac{n}2$-element subsets of $X$, one can partition $X$ into $\frac{n}2$ disjoint pairs in such a way that no matter how we pick one element from each of the first $\frac{n}2 - 1$ pairs, the set formed by them can always be completed to a member of $\mathcal{F}$ by adding an element of the last pair. The above problem is related to classical questions in extremal set theory. For any $t\ge 2$, we call a family of sets $\mathcal{F}\subset 2^X$ {\em $t$-separable} if for any ordered pair of elements $(x,y)$ of $X$, there exists $F\in\mathcal{F}$ such that $F\cap\{x,y\}=\{x\}$. For a fixed $t, 2\le t\le 5$ and $n\rightarrow\infty$, we establish asymptotically tight estimates for the smallest integer $s=s(n,t)$ such that every family $\mathcal{F}$ with $|\mathcal{F}|\ge s$ is $t$-separable.
