论文标题
在Kneser图的交点密度上$ K（n，3）$
On the intersection density of the Kneser Graph $K(n,3)$
论文作者
论文摘要
A集$ \ MATHCAL {F} \ subset \ operatorname {sym}（v）$是\ textsl {Intersecting}，如果其两个元素在$ v $的某些元素上一致。给定有限的传递排列组$ g \ leq \ peratatOrName {sym}（v）$，\ textsl {互动密度} $ρ（g）$是最大比率$ $ \ frac {| \ mathcal {| \ nathcal {f} \ textsl {交点密度} $ρ（x）$的tertex-transitive Graph $ x =（v，e）$等于$ \ max \ max \ max \ left \ weft \ {ρ（g）：g \ leq \ leq \ leq \ leq \ propatatorName {aut}（aut}（aut}（aut}（aut}（aut））在本文中，我们研究了$ n \ geq 7 $的neser图$ k（n，3）$的相交密度。 每当$ k（n，3）$的交叉密度在其自动形态组包含$ \ operatatorName {psl} _ {2}（q）$时确定，并取决于$ q $的一致性。我们还简要考虑$ k（n，2）$的交点密度为$ n $，其中$ \ operatatorName {psl} _ {2} _ {2}（q）$是其自动形态组的子组。
A set $\mathcal{F} \subset \operatorname{Sym}(V)$ is \textsl{intersecting} if any two of its elements agree on some element of $V$. Given a finite transitive permutation group $G\leq \operatorname{Sym}(V)$, the \textsl{intersection density} $ρ(G)$ is the maximum ratio $\frac{|\mathcal{F}||V|}{|G|}$ where $\mathcal{F}$ runs through all intersecting sets of $G$. The \textsl{intersection density} $ρ(X)$ of a vertex-transitive graph $X = (V,E)$ is equal to $\max \left\{ ρ(G) : G \leq \operatorname{Aut}(X), \mbox{ $G$ transitive} \right\}$. In this paper, we study the intersection density of the Kneser graph $K(n,3)$, for $n\geq 7$. The intersection density of $K(n,3)$ is determined whenever its automorphism group contains $\operatorname{PSL}_{2}(q)$, with some exceptional cases depending on the congruence of $q$. We also briefly consider the intersection density of $K(n,2)$ for values of $n$ where $\operatorname{PSL}_{2}(q)$ is a subgroup of its automorphism group.
