The Lovász Sieve

Abstract

Let A ₁, . . ., A _n be events in a probability space. In combinatorial applications the A _i are usually “bad” events. We wish to show Prob (∩ Ā _i ) > 0, so that there is a point (coloring, tournament, etc) which is “good.” By the counting sieve, this holds if Σ Prob (A _i ) < 1. In one sense this is best possible: when events A _i are pairwise disjoint, the condition cannot be weakened. At the opposite extreme, if the events A _i are mutually independent, then the only thing we need to ensure ∩ Ā _i ≠ ∅ is that all Prob (A _i ) < 1. The Lovász sieve is employed when there is “much independence” among the A _i .