  A \textit{linear $3$-graph}, $H = (V, E)$, is a set, $V$, of vertices
together with a set, $E$, of $3$-element subsets of $V$, called edges, so that
any two distinct edges intersect in at most one vertex. The linear Tur\'an
number, ${\rm ex}(n,F)$, is the maximum number of edges in a linear $3$-graph
$H$ with $n$ vertices containing no copy of $F$.
  We focus here on the \textit{crown}, $C$, which consists of three pairwise
disjoint edges (jewels) and a fourth edge (base) which intersects all of the
jewels. Our main result is that every linear $3$-graph with minimum degree at
least $4$ contains a crown. This is not true if $4$ is replaced by $3$. In fact
the known bounds of the Tur\'an number are \[ 6 \left\lfloor{\frac{n -
3}{4}}\right\rfloor \leq {\rm ex}(n, C) \leq 2n, \] and in the construction
providing the lower bound all but three vertices have degree $3$. We conjecture
that ${\rm ex}(n, C) \sim \frac{3n}{2}$ but even if this were known it would
not imply our main result.
  Our second result is a step towards a possible proof of ${\rm ex}(n,C) \leq
\frac{3n}{2}$ (i.e., determining it within a constant error). We show that a
minimal counterexample to this statement must contain certain configurations
with $9$ edges and we conjecture that all of them lead to contradiction.
