  There is a natural way to associate with a poset $P$ a hypergraph $H$, called
the hypergraph of critical pairs, so that the dimension of $P$ is exactly equal
to the chromatic number of $H$. The edges of $H$ have variable sizes, but it is
of interest to consider the graph $G$ formed by the edges of $H$ that have
size~2. The chromatic number of $G$ is less than or equal to the dimension of
$P$ and the difference between the two values can be arbitrarily large.
Nevertheless, there are important instances where the two parameters are the
same, and we study one of these in this paper. Our focus is on a family
$\{S_n^k:n\ge 3, k\ge 0\}$ of height two posets called crowns. We show that the
chromatic number of the graph $G_n^k$ of critical pairs of the crown $S_n^k$ is
the same as the dimension of $S_n^k$, which is known to be $\lceil
2(n+k)/(k+2)\rceil$. In fact, this theorem follows as an immediate corollary to
the stronger result: The independence number of $G_n^k$ is $(k+1)(k+2)/2$. We
obtain this theorem as part of a comprehensive analysis of independent sets in
$G_n^k$ including the determination of the second largest size among the
maximal independent sets, both the reversible and non-reversible types.
