  Given a finite point set $P$ in the plane, a subset $S \subseteq P$ is called
an island in $P$ if $conv(S) \cap P = S$. We say that $S\subset P$ is a visible
island if the points in $S$ are pairwise visible and $S$ is an island in $P$.
The famous Big-line Big-clique Conjecture states that for any $k \geq 3$ and
$\ell \geq 4$, there is an integer $n = n(k,\ell)$, such that every finite set
of at least $n$ points in the plane contains $\ell$ collinear points or $k$
pairwise visible points. In this paper, we show that this conjecture is false
for visible islands, by replacing each point in a Horton set by a triple of
collinear points. Hence, there are arbitrarily large finite point sets in the
plane with no 4 collinear members and no visible island of size $13$.
