Lecture 2. The Lovász Local Lemma

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1 Staford Uiversity Sprig 208 Math 233A: No-costructive methods i combiatorics Istructor: Ja Vodrák Lecture date: Jauary 0, 208 Origial scribe: Apoorva Khare Lecture 2. The Lovász Local Lemma 2. Itroductio ad motivatio We start with the Lovász Local Lemma, a fudametal tool of the probabilistic method ad a prototypical o-costructive argumet i combiatorics provig that a certai object exists without showig what it looks like. Ofte i applyig the probabilistic method, oe is tryig to show that it is possible to avoid bad evets E,..., E with positive probability, or i other words, [ ] P > 0. i= Here are subsets of a probability space Ω (typically a fiite set), ad = Ω \ deotes the complemetary evet for each i. If i P[] <, the the above iequality clearly follows, by the uio boud. However, this is ofte ot a strog eough tool, sice the sum i P[] may be much larger tha eve if the evets ca be avoided. A weaker costrait o the idividual probabilities P[ ] is sufficiet if the evets are also idepedet. I that case if P[ ] < for all i, the P[ i ] is clearly positive. The Lovász Local Lemma is a effective refiemet of this pheomeo, for evets that do ot have too much (iter)depedecy a otio that will be made precise presetly. A additioal attractive feature of the Local Lemma is that it does ot place ay restrictio o the (fiite) umber of evets. 2.2 Symmetric Local Lemma ad applicatio to hypergraph colorability Before we state ad prove the Local Lemma, we first preset a prototypical applicatio of the result, which serves to motivate it. Example 2. (Hypergraph 2-colorig) Give a iteger k 2, a k-uiform hypergraph G = (V (G), E(G)) cosists of a fiite set of odes V (G) ad a collectio of subsets e,..., e V (G), each of size k, which are termed edges (or hyper-edges ). We wat to color each ode i V (G) either red or blue. Uder what coditios ca we guaratee that there is a colorig with o moochromatic edge, i.e., every edge cotais both red ad blue odes? Such hypergraphs are said to be 2-colorable. Notice, if we color each ode red or blue uiformly at radom, the the evet that the ith edge is moochromatic has probability 2 k. Thus if the hypergraph G has less tha 2 k edges, the by the uio boud, the probability that there is at least oe moochromatic edge is < 2 k 2 k =. It follows that G is 2-colorable. However, this argumet fails whe G has 2 k edges. I this case, uder what assumptios ca we prove 2-colorability? Oe such assumptio is that every edge itersects at most d other

2 edges, for some d. Uder such a assumptio, we will show 2-colorability usig the Lovász Local Lemma. (Iterestigly, d will be comparable to 2 k.) We ow defie the followig otio of mutual idepedece. Defiitio 2.2 For all itegers > 0, defie [] := {,..., }. Give evets E,..., E Ω ad a subset J [], the evet is said to be mutually idepedet of {E j : j J} if for all choices of disjoit subsets J, J 2 J, P E j E j2 = P[ ] P E j E j2. j J j 2 J 2 j J j 2 J 2 Equipped with this otio, we ca state the first form of the Lovász Local Lemma, which will help aswer the above questio of 2-colorability for k-uiform hypergraphs. Theorem 2.3 (Symmetric Lovász Local Lemma) Suppose p (0, ), d, ad E,..., E are evets such that P[ ] p for all i. If each is mutually idepedet of all but d other evets E j, ad ep(d + ), where e = is Euler s umber, the P [ i= ] > 0. Remark 2.4 I the above result, d is sometimes called the depedece degree. The local -ess of the result has to do with the fact that assumptios deped oly o d rather tha, the umber of evets. Before we prove the Local Lemma (i a more geeral form), let us see how it ca be used to study 2-colorability for hypergraphs. I the settig of Example 2., suppose the hypergraph G has edges, deoted by e,..., e. Let deote the evet that the edge e i is moochromatic; as computed above, p = 2 k. We ow claim that d = d, that is, the d i Example 2. is precisely the d i Theorem 2.3. Ideed, fix a edge e i ; ow ay coditioig (i.e., ode-colorig) o the edges disjoit from e i is idepedet of, sice the ode colors are i.i.d. Beroulli radom variables. Thus, the assumptios of the Symmetric Lovász Local Lemma are ideed satisfied, as log as d + ep = 2k e. We stress agai that this coditio is idepedet of the umber of edges i the hypergraph G. 2.3 (Asymmetric) Lovász Local Lemma: statemet ad proof We ow prove the Symmetric Lovász Local Lemma, i.e., Theorem 2.3. I fact we show a stroger, asymmetric versio, ad use it to prove the symmetric versio. This will require the followig useful cocept. Defiitio 2.5 A (directed) graph G = (V (G), E(G)) is a depedecy (di)graph o evets E,..., E if V (G) = [] ad each evet is mutually idepedet of its o-eighbors {E j : j i, (i, j) E(G)}.

3 Remark 2.6 Most applicatios i the literature use the udirected versio of the depedecy graph; however, there are some applicatios that use the digraph structure. I such cases, give a directed edge (i, j), i is the source ad j the target. We ca ow state the Lovász Local Lemma i its more geeral form. Theorem 2.7 ((Asymmetric) Lovász Local Lemma) Suppose G is a depedecy (di)graph for evets E,..., E, ad there exist x,..., x (0, ) such that P[ ] x i (i,j) E(G) ( x j ), i []. (2.) The, [ ] P ( x i ) > 0. (2.2) i= i= Remark 2.8 Give a set of evets, the choice of a depedecy digraph G is ot uique, or is the choice of the parameters x i. Rather, the user decides which depedecy digraph G ad parameters x i to work with, i a give applicatio. The depedecy graph is ofte clear from the cotext (e.g. i the hypergraph colorability applicatio above), although the choice of x i might ot be. Remark 2.9 Theorem 2.7 is sharp whe the are idepedet, G is empty, ad x i = P[ ] i. Before we show the Asymmetric Local Lemma, let us quickly see why it implies the Symmetric versio. Ideed, if the hypotheses of Theorem 2.3 hold, set x i = d+ i. Now the hypotheses imply that there is a udirected depedecy graph G i which each ode has degree at most d. Therefore, x i (i,j) E(G) ( x j ) = d + ( ) deg(i) ( ) d d + d + d + It follows by the Asymmetric Lovász Local Lemma that P[ i ] > 0. Fially, we prove the Asymmetric Lovász Local Lemma. Proof of Theorem 2.7. Give S [], defie [ ] P S := P, P :=. i S d + e P[]. The result follows oce we show, by iductio o S, that for all S [] ad a S, P S x a. (2.3)

4 More precisely, we will show by iductio o S that P S ( x a ) > 0. Ideed, this yields the result, because applyig the iequality to S = [], the [ ], ad so o, yields: [ ] P = P [] ( x )P [ ] = ( x )( x )P [ 2] ( x i ) > 0, i= as desired. Thus it remais to prove (2.3). The base case is whe S = {a} is a sigleto. I this case, P {a} P = P[E a ] x a (a,j) E(G) ( x j ) x a, provig the assertio. Now suppose (2.3) holds for all subsets S [] with size at most k, ad say S [] has size k +. To proceed further, let us defie the eighborhood of a S, as well as its closure, via: Now fix a S, ad compute: [ ] P S = P = P i S Γ(a) := {j V (G) : (a, j) E(G)}, Γ + (a) := {a} Γ(a). (2.4) i S\{a} P E a i S\{a} P i S\{a} i= P E a = P[E a ]P S\Γ + (a), i S\Γ + (a) where the first equality ad the iequality are straightforward, ad the fial equality follows from the mutual idepedece of E a ad { : i Γ + (a)}. From this computatio it follows that P S P[E a ] P S\Γ+ (a), where > 0 by the iductio hypothesis. Now say Γ(a) S = {b,..., b d } for some d 0, ad write the fractio o the right-had side as a telescopig product: P S\Γ + (a) = P S\{a,b } P S\{a,b,b 2 } P S\{a,b } P S\{a,b,...,b d } P S\{a,b,...,b d } where all terms o the right-had side are strictly positive by the iductio hypothesis. By the same hypothesis, each ratio o the right-had side is bouded above by. Therefore, x bi P S\Γ + (a). x b x bd Recallig that by assumptio P[E a ] x a b Γ(a) ( x b), it follows that P S x a b Γ(a) ( x b ) c Γ(a) S This shows (2.3), ad with it, the Lovász Local Lemma. x c x a > 0.,

5 3 Applicatio to Ramsey umbers As a applicatio of the asymmetric LLL (Theorem 2.8) we will prove a lower boud for Ramsey umbers R(3,. For itegers k, l 2 the Ramsey umber R(k, is the miimal positive iteger N, such that for every edge-colorig of the complete graph K N there exists a red clique of size k or a blue clique of size l. By iductio o k ad l, oe ca easily prove a upper boud ( ) k + l 2 R(k,. k We will ow cosider the case k = 3 ad l 3. The the above iequality reads R(3, 2l(l + ). We would like obtai a lower boud for R(3, from the LLL. I order to do so, we cosider a complete graph o vertices ad a radom colorig of its edges with the colors red ad blue: We color each edge red with probability p ad blue with probability p; idepedetly for all edges. Here the umber of vertices ad the probability p will be specified later (depedig o the value of. Our goal is to obtai with positive probability a colorig without a red triagle ad without a blue l-clique, sice this would establish the lower boud R(3, >. For each 3-elemet subset T of the vertex set let A T be the evet that the three vertices i T form a red triagle. Note that for each such T we have P(A T ) = p 3 ad the umber of these evets A T is ( 3). Furthermore, for each l-elemet subset S of the vertex set let BS be the evet that the l vertices i S form a blue clique. Note that for each such S we have P(B S ) = ( p) (l 2) ad the umber of these evets B S is (. Let us ow defie a depedecy graph for these evets. We joi two evets of the form A T or B S, if the correspodig sets S or T itersect i at least two vertices (i.e. if they share a edge). It is clear that a evet A T or B S is mutually idepedet of all evets with which it does ot share a edge. Hece the graph we just defied is ideed a depedecy graph. Let us ow boud the degrees i this graph. First cosider a vertex A T. It is coected to at most 3 evets A T. (I order to be coected, the triagles T ad T eed to share a edge. T has 3 edges ad for each of them there are at most choices of a third vertex to form a triagle T ). Also, trivially A T is coected to at most ( evets BS. (We use a trivial boud here sice the actual umber of depedecies is ot much less.) Now, let us cosider a vertex B S. It is coected to at most ( l 2) evets AT (I order to have a coectio, the sets S ad T eed to itersect i at least 2 elemets. There are ( l 2) choices of two elemets i S ad for each of them at most choices for the third vertex i T ). Also, trivially B S is coected to at most ( evets BS. So i order to apply the LLL we eed to fid positive real umbers x, y (0, ) with p 3 = P(A T ) x( x) 3 ( y) ( ad ( p) (l 2) = P(BS ) y( x) (l 2) ( y) (.

6 Here x ad y shall be the values x i i Theorem 2.8. Note that we choose the same value x for all evets A T (sice the situatio is symmetric for all choices of T ) ad the same value y for all evets B S (sice the situatio is symmetric for all choices of S). Let us assume that for some value of we ca fid positive real umbers p, x, y (0, ) such that both of the above iequalities are fulfilled. The the LLL yields that with positive probability oe of the evets A T ad B S happe. This meas that there is a colorig of the edges of a complete graph o vertices such that there is o red triagle ad o blue l-clique. Therefore we have R(3, >, if we ca fid p, x, y (0, ) as above. Let us ow try to fid such p, x, y (0, ) for as large as we ca. We guess that y = ( would be a smart choice, because the the term ( y) ( is roughly costat (aroud e ). Furthermore, we observe that p ad x eed to fulfill the followig iequalities: ad p 3 x( x) 3 ( y) ( x e p(l 2) ( p) ( l 2) y( x) ( l 2) ( y) ( ( x) ( l 2) e x(l 2). Hece we eed p x p 3 ad therefore p. Fially, to guess the depedece of o l we ote e p(l 2) ( p) ( 2) l y( x) ( 2) l ( y) ( y = ( e l log, hece pl 2 p ( ) l 2 l log ad therefore l p log log. Motivated by this we assume l 20 log ad choose y = (, x = ad p = 9 3/2 3 (the costats here are ot really importat, they are just chose i such a way that the iequalities work out). The we have ( ) ( ( y) ( = ( e.0 if is sufficietly large (sice ( m )m e as m ). Furthermore for sufficietly large we also have ( ( x) 3 = ) 3 9 3/2 3 e 0.0 for sufficietly large. Thus, p 3 = 27 3/2 9 3/2 e.02 x( x) 3 ( y) (, which establishes the first desired iequality. For the secod iequality ote that for sufficietly small h > 0 we have h e 2h. So for sufficietly large we get ( x) (l 2) e 2x(l 2) = e 2 9 (l 2). Furthermore, usig l 20 log, y = ( ) l l = e l log e l2 20 e l(l ) 9 e l(l ) = e 9 (l 2)+.0.

7 Hece ( p) (l 2) e p( l 2) = e 3 (l 2) = e 9 (l 2)+.0 e 2 9 (l 2) e.0 y( x) (l 2) ( y) (, which verifies the secod desired iequality. Thus, for l 20 log we ca fid p, x, y (0, ) with the two desired iequalities. This implies (by the LLL, as described above) that R(3, > wheever l 20 log ad sufficietly large. Note that l 2 (40 log 2 implies 20 l log log l log(l2 ) = l. Therefore we have R(3, l 2 (40 log 2 for all sufficietly large l. This proves the followig theorem. Theorem 3. There is a positive costat c > 0, such that R(3, c l2 log 2 l for all l 3. So we have proved c l2 log 2 l R(3, 2l(l + ). We coclude this sectio by remarkig that the ) true aswer is R(3, = Θ. (The upper boud was proved by Ajtai, Komlós, Szemerédi ad ( l 2 log l the lower boud by Jeog Ha Kim.)