论文标题
卡托纳交叉点定理的本地版本
A local version of Katona's intersection theorem
论文作者
论文摘要
Katona的交集定理指出，每个相交的家庭$ \ Mathcal f \ subseteq [n]^{（k）} $满足$ \ vert \ pertial \ partial \ partcal \ nathcal f \ vert \ geq \ geq \ vert \ vert \ vert \ mathcal f \ vert f \ vert $ f \ in \ Mathcal f \} $是$ \ Mathcal f $的阴影。 Frankl猜想，对于$ n> 2k $，每个相交的家庭$ \ Mathcal f \ subseteq [n]^{（k）} $，在[n] $中有一些$ i \，以至于$ \ vert \ vert \ partial \ partial \ partcal f（i） f（i）= \ {f \ setminus i：i \ in f \ in \ mathcal f \} $是$ i $的$ \ MATHCAL f $的链接。在这里，我们以$ n> \ binom {k+1} {2} $的强烈形式证明了这种猜想。特别是，我们的结果意味着对于[k] $中的任何$ j \，都有一个$ j $ -set $ \ {a_1，\ dots，a_j \} \ in [n]^{（j）} $，因此$ \ vert \ vert \ vert \ partial \ partial \ natercal f（a_1，\ dots，a_1，\ dots，a_________q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q QQ fert f（a_1，\ dots，a_j）\ vert $。也为跨交流家庭获得了类似的陈述。
Katona's intersection theorem states that every intersecting family $\mathcal F\subseteq[n]^{(k)}$ satisfies $\vert\partial\mathcal F\vert\geq\vert\mathcal F\vert$, where $\partial\mathcal F=\{F\setminus x:x\in F\in\mathcal F\}$ is the shadow of $\mathcal F$. Frankl conjectured that for $n>2k$ and every intersecting family $\mathcal F\subseteq [n]^{(k)}$, there is some $i\in[n]$ such that $\vert \partial \mathcal F(i)\vert\geq \vert\mathcal F(i)\vert$, where $\mathcal F(i)=\{F\setminus i:i\in F\in\mathcal F\}$ is the link of $\mathcal F$ at $i$. Here, we prove this conjecture in a very strong form for $n> \binom{k+1}{2}$. In particular, our result implies that for any $j\in[k]$, there is a $j$-set $\{a_1,\dots,a_j\}\in[n]^{(j)}$ such that $\vert \partial \mathcal F(a_1,\dots,a_j)\vert\geq \vert\mathcal F(a_1,\dots,a_j)\vert$. A similar statement is also obtained for cross-intersecting families.
