MTH 742 P : Advanced Combinatorics 28 / 11 / 2016 Lecture 9 : Alteration Lecturer :

Sometimes merely randomization is not enough to show what we want. A common strategy is then to alter the instance we construct via randomness. For example, if the expectation of the parameter X of a random object is EX, then we have an instance with parameter ≤ EX. However, what we are after might be a “perfect” object with X = 0. Thus, what we do is to “correct bad properties”. Once again, this is best shown by examples. An independent set of a graph is the complement of a clique; that is, S is an independent set, if there is no edge between vertices of S.