  We prove two theorems concerning incidence posets of graphs, cover graphs of
posets and a related graph parameter. First, answering a question of Haxell, we
show that the chromatic number of a graph is not bounded in terms of the
dimension of its incidence poset, provided the dimension is at least four.
Second, answering a question of K\v{r}\'{i}\v{z} and Ne\v{s}et\v{r}il, we show
that there are graphs with large girth and large chromatic number among the
class of graphs having eye parameter at most two.
