  Let $S$ be an $n$-punctured sphere, with $n \geq 3$. We prove that
$\binom{n}{3}$ is the maximum size of a family of pairwise non-homotopic simple
arcs on $S$ joining a fixed pair of distinct punctures of $S$ and pairwise
intersecting at most twice. On the way, we show that a square annular diagram
$A$ has a corner on each of its boundary paths if $A$ contains at least one
square and the dual curves of $A$ are simple arcs joining the boundary paths of
$A$ and pairwise intersecting at most once.
