More on random graphs and discrepancy for directed graphs

Consider the following problem: Does there exist a graph with large girth and large chromatic number? The girth of a graph is the length of the shortest cycle and the chromatic number is the minimum number of colors needed to color the vertices of the graph so no two adjacent vertices have the same color. These two conditions seem contradictory, intuitively to have high chromatic number we need something dense, i.e., clusters of vertices all close to one another. On the other hand to have large girth means that locally everything looks like a tree (i.e., no local cycles) and hence locally the graph is 2-colorable. An amazing result of Erdős is that such graphs exist with girth at least g and chromatic number at least k for any positive g and k. We now give a proof using random graphs (this proof is from [4]).