Barcelona Graduate School of Mathematics, Fall 2013
Random Structures and the Probabilistic Method
Session 3: The Lovász local lemma
1. The local lemma
Let A1 , . . . , An be events in a probability space. A dependency graph D = (V, E) for this set of events is a
directed graph with V = [1, n] such that each Ai is jointly independent with every set of events {Aj : j ∈ J}
for which J ∩ N [i] = ∅, where N [i] denotes the close out–neighborhood of i in D: Pr(Ai | ∪j∈J Aj ) = Pr(Ai )
for each i. The following lemma appeared ina a paper by Erdős and Lovász in 1978.
Theorem 1.1 (Lovász local lemma). Let A1 , . . . , An be events in a probability space and let D = (V, E) be
a dependency graph for this set of events. Let x1 , . . . , xn ∈ (0, 1). If
Y
Pr(Ai ) ≤ xi
(1 − xj ), i = 1, . . . , n,
j:(i,j)∈E
then
Pr(∩ni=1 Āi ) ≥
n
Y
(1 − xi ).
i=1
In applications the simpler symmetric version usually suffices.
Corollary 1.2 (Symmetric version). Let A1 , . . . , An be events in a probability space and let D = (V, E) be
a dependency graph for this set of events. Let d be the maximum out–degree of D. If
Pr(Ai ) ≤ p, i = 1, . . . , n, and ep(d + 1) < 1,
then
Pr(∩ni=1 Āi ) ≥ (1 − 1/(d + 1))n .
2. Some classical applications
Theorem 2.1 (improved lower bound on van der Waerden number). The van der Waerden number W (k)
for two colors satisfies W (k) ≥ 2k /8k.
A multidimensional extension of van der Waerden theorem due to Gallai states that, for every finite set
S and every col9oring of Rd with a finite number of colors, there is an homothetic copy aS + b of S which
is monochromatic. A related result for multicolroed sets is the following.
Theorem 2.2 (Erdős and Lóvasz, 1975). For each finite set S ⊂ R with m = |S| ≥ (3 + o(1))k log k there
is a coloring c : R → {1, . . . , k} of the real line such that every translation x + S of S contains all k colors.
A colection of open unit balls of R3 is a k–fold covering if every point is in at least k balls. It is decomposable
if it can be split into two coverings. There are indecomposable k–fold coverings of R3 , but the following
result says, somewhat surprisingly, that a local condition implies decomposability.
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Theorem 2.3 (Mani-Levitska-Pach, 19). Let F be a k–fold covering of R3 such that no point is in more
than t balls in F. If 218 et8 /2k−1 ≤ 1 then F is decomposable.
3. Exercises
(1) A hypergraph H = (V, E) is a pair V , a set of vertices, and E, simply a collection of subsets of
V . The hypergraph H is 2–colorable if there is a coloring of the vertices with two colors such that
no edge is monochromatic. Let H be a hypergraph in which each edge has at least k vertices and
intersects at most d edges. If e(d + 1) ≤ 2k−1 then H is 2–colorable. In particular every k–uniform
and k–regular hypergraph with k ≥ 9 is 2–colorable.
[Hint: color each vertex red or blue independently with the same probability. For each edge e let
Ae be the event that e is monochromatic. Apply the local lemma.]
(2) Let G = (V, E) be a graph. Let L = {L(v) ⊂ N : v ∈ V, } a family of sets each with ` = |L(v)|
elements. Suppose that, for each edge xy ∈ E we have |L(u) ∩ L(v)| ≤ `/8. Then there is a proper
coloring c : V → N such that c(v) ∈ L(v) for each v ∈ V .
[Hint: Consider a random coloring in which every vertex v is assigned an element in L(v) with
uniform distribution. For each edge xy ∈ E and every color i ∈ L(x) ∩ L(y) define Axy,i the event
that c(x) = c(y). Apply the local lemma.]
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