Incidence Posets and Cover Graphs

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?í? and Ne?et?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.