Blocking Amalgamations, Maximal Arcs, and Generalized Crowns

Abstract

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.