阻塞合并、极大弧与广义皇冠

中文摘要 AI 辅助

设$C^r_{1,k}$为$r$一致$k$-皇冠，令$h=r-k+2$。对于有限线性相交$r$一致超图$G$，定义$\tau_h(G)$为满足与$G$每条边相交至少$h$个顶点的最小集合大小，且$\rho_{r,k}=\frac{|E(G)|}{\tau_h(G)}$的上确界。我们证明，对任意固定对$(G,B)$（其中$B$为$h$重横截），当$n$足够大时，线性$r$一致超图中不含$C^r_{1,k}$的极值边数$\text{ex}^{\text{lin}}_r(n,C^r_{1,k})\frac{|E(G)|}{|B|}n-O_{G,B}(\trianglelefteq n)$。关联计数给出$\rho_{r,k}\frac{r}{h}$，且经对偶化后，当存在带特殊正则子族的成对平衡设计时等号成立。当$r=q+1$（$q$为素数幂）时，截断射影平面给出$\frac{q}{h}\rho_{q+1,k}\frac{q+1}{h}$；若存在极大$h$-弧，则达到上界，特别地，当$q$为偶数且$h\backslash q$时，$\rho_{q+1,k}=\frac{q+1}{h}$。对截断平面构造进行填充可得$\rho_{r,r}=(1-o(1))\frac{r}{2}$，且对任意固定$\trianglelefteq0$及$\trianglelefteq r k r$，$\rho_{r,k}=(1+o(1))\frac{r}{r-k+2}$。对于非相交模板，对应的转换由局部安全块条件控制，该条件替代了$h$重横截要求。

英文摘要

Let $C^r_{1,k}$ be the $r$-uniform $k$-crown and put $h=r-k+2$. For a finite linear intersecting $r$-uniform hypergraph $G$, let $τ_h(G)$ be the minimum size of a set meeting every edge of $G$ in at least $h$ vertices, and define \[ ρ_{r,k}=\sup_G\frac{|E(G)|}{τ_h(G)}. \] We prove that every fixed pair $(G,B)$, with $B$ an $h$-fold transversal, yields \[ \operatorname{ex}^{\mathrm{lin}}_r(n,C^r_{1,k}) \ge \frac{|E(G)|}{|B|}n-O_{G,B}(\sqrt n) \] for all sufficiently large $n$. Incidence counting gives $ρ_{r,k}\le r/h$, and equality is characterized after dualization by a pairwise balanced design with a distinguished regular subfamily. For $r=q+1$, where $q$ is a prime power, truncated projective planes give \[ \frac qh\le ρ_{q+1,k}\le\frac{q+1}{h}. \] The upper endpoint is attained whenever a maximal $h$-arc exists; in particular, if $q$ is even and $h\mid q$, then $ρ_{q+1,k}=(q+1)/h$. Padding the truncated-plane construction gives \[ ρ_{r,r}=(1-o(1))\frac r2 \] and, uniformly for each fixed $\varepsilon>0$ and $\varepsilon r\le k\le r$, \[ ρ_{r,k}=(1+o(1))\frac{r}{r-k+2}. \] For nonintersecting templates, the corresponding transfer is governed by a local safe-block condition that replaces the $h$-fold transversal requirement.

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17 pages