  The original notion of dimension for posets is due to Dushnik and Miller and
has been studied extensively in the literature. Quite recently, there has been
considerable interest in two variations of dimension known as Boolean dimension
and local dimension. For a poset $P$, the Boolean dimension of $P$ and the
local dimension of $P$ are both bounded from above by the dimension of $P$ and
can be considerably less. Our primary goal will be to study analogies and
contrasts among these three parameters. As one example, it is known that the
dimension of a poset is bounded as a function of its height and the tree-width
of its cover graph. The Boolean dimension of a poset is bounded in terms of the
tree-width of its cover graph, independent of its height. We show that the
local dimension of a poset cannot be bounded in terms of the tree-width of its
cover graph, independent of height. We also prove that the local dimension of a
poset is bounded in terms of the path-width of its cover graph. In several of
our results, Ramsey theoretic methods will be applied.
