LTCC Course: Graph Theory January-February 2011 §4 Probabilistic Methods and Random Graphs

Is there, for all large values of n, a graph on n vertices with no clique of size 5 log n, and no independent set of size 5 log n? Is there a graph with some number n of vertices, no cycles of length less than 100, and no independent sets of size greater than n/100? Note that such a graph has chromatic number at least 100. Is there, for all large values of n, a graph on n vertices with maximum degree at most n2/3, and, for every pair (U, V ) of sets of vertices with |U |, |V | ≥ n1/2, there is an edge from U to V ? The answer to each of these questions is yes, but how might one go about proving that? The natural first reaction on being confronted by questions like these is to try and construct graphs with the required combination of properties. In each case, one of the two required properties demands that the graph have rather few edges, and the other demands rather many. It’s hard to