论文标题
$ 3 $ - 对称组的相交家庭
$3$-setwise intersecting families of the symmetric group
论文作者
论文摘要
给定两个正整数$ n \ geq 3 $和$ t \ leq n $，\ operatorname {sym}（sym}（n）$的排列$σ，π\是$ t $ - set的相交，如果他们同意（setwise）在$ t $ -subset of $ t $ -subset of $ \ subs of $ \ \ \ \ \ {1,2，\ {1,2，\ ldots $} $。一个家庭$ \ MATHCAL {F} \ subset \ operatorName {sym}（n）$是$ t $ - setwise相交，如果$ \ mathcal {f} $的两个排列是$ t $ t $ - setwise的相交。 Ellis [组合理论杂志，第A系，119（4），825--849，2012]指出，如果$ t \ leq n $和$ \ Mathcal {f} \ subset \ subset \ subset \ operatotorname {sym} {sym}（sym}（sym}（n）$仅当$ \ Mathcal {f} $是$ t $ -subset的$ \ {1,2，\ ldots，n \} $的固定稳定器的固定时。 在本文中，我们证明，如果$ n \ geq 11 $和$ \ mathcal {f} $是$ 3 $ -SetWise Intersecting，则$ | \ Mathcal {f} | \ leq 6（n-3）！$。此外，我们证明，最大尺寸的$ 3 $ - set的相交家族的特征向量在于由$ \ operatotorname {sym}（sym}（n）$ of $ 3 $ -subset of $ 3 $ -subsets of $ \ \ \ {1,2，\ ldots，\ ldots，n \ \} $的特征。
Given two positive integers $n\geq 3$ and $t\leq n$, the permutations $σ,π\in \operatorname{Sym}(n)$ are $t$-setwise intersecting if they agree (setwise) on a $t$-subset of $\{1,2,\ldots,n\}$. A family $\mathcal{F} \subset \operatorname{Sym}(n)$ is $t$-setwise intersecting if any two permutations of $\mathcal{F}$ are $t$-setwise intersecting. Ellis [Journal of Combinatorial Theory, Series A, 119(4), 825--849, 2012] conjectured that if $t\leq n$ and $\mathcal{F} \subset \operatorname{Sym}(n)$ is a $t$-setwise intersecting family, then $|\mathcal{F}|\leq t!(n-t)!$ and equality holds only if $\mathcal{F}$ is a coset of a setwise stablizer of a $t$-subset of $\{1,2,\ldots,n\}$. In this paper, we prove that if $n\geq 11$ and $\mathcal{F}$ is $3$-setwise intersecting, then $|\mathcal{F}|\leq 6(n-3)!$. Moreover, we prove that the characteristic vector of a $3$-setwise intersecting family of maximum size lies in the sum of the eigenspaces induced by the permutation module of $\operatorname{Sym}(n)$ acting on the $3$-subsets of $\{1,2,\ldots,n\}$.
