  This paper studies problems related to visibility among points in the plane.
A point $x$ \emph{blocks} two points $v$ and $w$ if $x$ is in the interior of
the line segment $\bar{vw}$. A set of points $P$ is \emph{$k$-blocked} if each
point in $P$ is assigned one of $k$ colours, such that distinct points $v,w\in
P$ are assigned the same colour if and only if some other point in $P$ blocks
$v$ and $w$. The focus of this paper is the conjecture that each $k$-blocked
set has bounded size (as a function of $k$). Results in the literature imply
that every 2-blocked set has at most 3 points, and every 3-blocked set has at
most 6 points. We prove that every 4-blocked set has at most 12 points, and
that this bound is tight. In fact, we characterise all sets
$\{n_1,n_2,n_3,n_4\}$ such that some 4-blocked set has exactly $n_i$ points in
the $i$-th colour class. Amongst other results, for infinitely many values of
$k$, we construct $k$-blocked sets with $k^{1.79...}$ points.
