LTCC Course: Graph Theory January-February 2013 §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? ∗These notes are based on Graham Brightwell’s notes for LTCC Graph Theory Courses in 2009-12.