论文标题
与家族相交的三角形
Triangles in intersecting families
论文作者
论文摘要
我们证明了以下广义的Turán类型结果。 $ r $ sets的$ r $ $ sets $ r $ triangle的集合是一个$ r $ - 三角\ Mathcal {t}} t $是空的。集合的家庭$ \ Mathcal {f} $是$ r $ - 如果任何$ f_1，f_1，f_2，\ dots，f_r \ in \ nathcal {f} $，我们有$ \ cap_ {i = 1}^rf_i \ neq \ neyq \ emertyt $或等于$ n $ nire $ n $ nir $ n $ nir $ nir $ {对于$ m = 2,3，\ dots，r $。我们证明，如果$ n \ ge n_0（r，k）$，则$ r $ - 相交的家庭$ \ mathcal {f} \ subseteq \ binom {[n]} {k} {k} {k} {k} {k}包含最多$（r+1）$ - triangles to Isomorphic to Isomorphic to $ \ \ \ \ f \ \ binom {[n]} {k}：| f \ cap [r+1] | \ ge r \ \} $。
We prove the following the generalized Turán type result. A collection $\mathcal{T}$ of $r$ sets is an $r$-triangle if for every $T_1,T_2,\dots,T_{r-1}\in \mathcal{T}$ we have $\cap_{i=1}^{r-1}T_i\neq\emptyset$, but $\cap_{T\in \mathcal{T}}T$ is empty. A family $\mathcal{F}$ of sets is $r$-wise intersecting if for any $F_1,F_2,\dots,F_r\in \mathcal{F}$ we have $\cap_{i=1}^rF_i\neq \emptyset$ or equivalently if $\mathcal{F}$ does not contain any $m$-triangle for $m=2,3,\dots,r$. We prove that if $n\ge n_0(r,k)$, then the $r$-wise intersecting family $\mathcal{F}\subseteq \binom{[n]}{k}$ containing the most number of $(r+1)$-triangles is isomorphic to $\{F\in \binom{[n]}{k}:|F\cap [r+1]|\ge r\}$.
