Blocking Amalgamations, Maximal Arcs, and Generalized Crowns

Let $C_{1,k}^r$ be the $r$-uniform $k$-crown and put $h=r-k+2$. For a finite linear intersecting $r$-uniform hypergraph $G$, let ${\tau }_h(G)$ be the minimum size of a set meeting every edge of $G$ in at least $h$ vertices, and define $${\rho }_{r,k}=\underset{G}{\sup }\frac{\mathrm{\mid }E(G)\mathrm{\mid }}{{\tau }_h(G)}\mathrm{.}$$ We prove that every fixed pair $(G,B)$, with $B$ an $h$-fold transversal, yields $${\mathrm{ex}}_r^{\mathrm{l}\mathrm{i}\mathrm{n}}(n,C_{1,k}^r)\ge \frac{\mathrm{\mid }E(G)\mathrm{\mid }}{\mathrm{\mid }B\mathrm{\mid }}n-O_{G,B}(\sqrt{n})$$ for all sufficiently large $n$. Incidence counting gives ${\rho }_{r,k}\le r\mathrm{/}h$, 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 $$\frac{q}{h}\le {\rho }_{q+1,k}\le \frac{q+1}{h}\mathrm{.}$$ The upper endpoint is attained whenever a maximal $h$-arc exists; in particular, if $q$ is even and $h\mid q$, then ${\rho }_{q+1,k}=(q+1)\mathrm{/}h$. Padding the truncated-plane construction gives $${\rho }_{r,r}=(1-o(1))\frac{r}{2}$$ and, uniformly for each fixed $\epsilon >0$ and $\epsilon r\le k\le r$, $${\rho }_{r,k}=(1+o(1))\frac{r}{r-k+2}\mathrm{.}$$ For nonintersecting templates, the corresponding transfer is governed by a local safe-block condition that replaces the $h$-fold transversal requirement.