Math 233 A : Non - constructive methods in combinatorics Instructor :

Here Ei are subsets of a probability space Ω (typically a finite set), and E i = Ω \ Ei denotes the complementary event for each i. If ∑ i P[Ei] < 1, then the above inequality clearly follows, by the “union bound”. However, this is often not a strong enough tool, since the sum ∑ i P[Ei] may be much larger than 1 even if the events can be avoided. A weaker constraint on the individual probabilities P[Ei] is sufficient if the events Ei are also independent. In that case if P[Ei] < 1 for all i, then P[∩iE i] is clearly positive. The Lovász Local Lemma is an effective refinement of this phenomenon, for events that do not have “too much (inter)dependency” – a notion that will be made precise presently. An additional attractive feature of the Local Lemma is that it does not place any restriction on the (finite) number of events Ei.