论文标题
最大的非平凡的D-Wise相交均匀族的结构较大
The structure of maximal non-trivial d-wise intersecting uniform families with large sizes
论文作者
论文摘要
对于一个正整数$ d \ geq 2 $，一个家庭$ \ Mathcal f \ subseteq \ binom {[n]} {k} {k} $被认为是d-wise相交的，如果$ | f_1 \ cap f_1 \ cap f_2 \ cap f_2 \ cap \ cap \ dots \ cap f_q f_q f_q 1 $ $ f_q for_ f _ $ f_1 $ f_ f _ f _ f_1 $ f_2 f $。一个d-wise互为$ \ mathcal f \ subseteq \ binom {[n]} {k} $如果$ \ mathcal f \ cup \ {a \} $不是d-wise Inspection d d-wise In \ in \ in \ in \ binom {binom {n n] $ a的$ a {我们提供了O'Neill和Verstraëte定理的改进，内容涉及最大和第二大最大非平凡的D-Wise Interpect k-均匀家庭的结构。我们还确定了第三大和第四大的最大非活力D-Wise与任何$ k> d+1 \ 1 \ geq 4 $相交的K-均匀家庭，以及第五大，第五大，第六大最大最大3-Wise 3-wise Intersect k-Strifting K-均匀的家族，以供任何$ k \ geq 5 $ \ geq 5 $，asmptity asmptity asmptity asmptity。我们的证明是$δ$系统方法的应用。
For a positive integer $d\geq 2$, a family $\mathcal F\subseteq \binom{[n]}{k}$ is said to be d-wise intersecting if $|F_1\cap F_2\cap \dots\cap F_d|\geq 1$ for all $F_1, F_2, \dots ,F_d\in \mathcal F$. A d-wise intersecting family $\mathcal F\subseteq \binom{[n]}{k}$ is called maximal if $\mathcal F\cup\{A\}$ is not d-wise intersecting for any $A\in\binom{[n]}{k}\setminus\mathcal F$. We provide a refinement of O'Neill and Verstraëte's Theorem about the structure of the largest and the second largest maximal non-trivial d-wise intersecting k-uniform families. We also determine the structure of the third largest and the fourth largest maximal non-trivial d-wise intersecting k-uniform families for any $k>d+1\geq 4$, and the fifth largest and the sixth largest maximal non-trivial 3-wise intersecting k-uniform families for any $k\geq 5$, in the asymptotic sense. Our proofs are applications of the $Δ$-system method.
