On the Probability of Independent Sets in Random Graphs

Transcription

1 O the Probability of Idepedet Sets i Radom Graphs Michael Krivelevich,, * Bey Sudakov, 2,3, VaH.Vu, 4, Nicholas C. Wormald 5, Departmet of Mathematics, Raymod ad Beverly Sackler Faculty of Exact Scieces, Tel Aviv Uiversity, Tel Aviv 69978, Israel; krivelev@post.tau.ac.il 2 Departmet of Mathematics, Priceto Uiversity, Priceto, New Jersey Istitute for Advaced Study, Priceto, New Jersey 08540; bsudakov@math.priceto.edu 4 Departmet of Mathematics, Uiversity of Califoria at Sa Diego, La Jolla, Califoria 92093; vavu@ucsd.edu 5 Departmet of Mathematics ad Statistics, Uiversity of Melboure VIC 300, Australia; ick@ms.uimelb.edu.au Received 4 February 200; accepted 2 September 2002 DOI 0.002/rsa.0063 ABSTRACT: Let k be the asymptotic value of the idepedece umber of the radom graph G(, p). We prove that if the edge probability p() satisfies p() 2/5 l 6/5 the the probability that G(, p) does ot cotai a idepedet set of size k c, for some absolute costat c 0, is at most exp{c 2 /(k 4 p)}. We also show that the obtaied expoet is tight up to logarithmic factors, ad apply our result to obtai ew bouds o the choice umber of radom Correspodece to: M. Krivelevich *Research supported i part by a USA-Israel BSF Grat, by a grat from the Israel Sciece Foudatio, ad by a Bergma Memorial Grat. Research supported i part by NSF Grats DMS , CCR , ad by the State of New Jersey. Research supported i part by Grat RB09G-Vu from UCSD, a Sloa Fellowship ad a NSF Grat. Research supported by the Australia Research Coucil Wiley Periodicals, Ic.

2 2 KRIVELEVICH ET AL. graphs. We also discuss a geeral settig where our approach ca be applied to provide a expoetial boud o the probability of certai evets i product probability spaces Wiley Periodicals, Ic. Radom Struct. Alg., 22: 4, INTRODUCTION Let G(, p) deote as usual the probability space whose poits are graphs o labeled vertices {,..., }, where each pair of vertices forms a edge radomly ad idepedetly with probability p p(). We say that the radom graph G(, p) possesses a graph property A asymptotically almost surely, or a.a.s. for short, if the probability that G(, p) satisfies A teds to as the umber of vertices teds to ifiity. Defie the followig quatity: k* maxk : k p k 2. I words, k* is the maximum iteger k for which the expectatio of the umber of idepedet sets of size k i G(, p) is still at least. It has bee kow for a log time [6, 3] that, for large eough p p() (say, for p() for small eough costat 0) a.a.s. i G(, p), the idepedece umber of G is asymptotically equal to k*. I fact, usig the so-called secod momet method, oe ca prove that uder the above assumptios the idepedece umber of G(, p) is cocetrated a.a.s. i two cosecutive values, oe of them beig k*. Now let us pick a iteger k 0 slightly less tha k* (we will be more precise later) ad ask the followig: What is the probability that the radom graph G(, p) does ot cotai a idepedet set of size k 0? This seemigly somewhat artificial questio turs out to be of extreme importace for may deep problems i the theory of radom graphs. A expoetial estimate of the above probability provided a crucial igrediet i the semial breakthrough of Bollobás [5], establishig the asymptotic value of the chromatic umber of radom graphs. Later, this problem became a fruitful playgroud for comparig the stregth of various large deviatio methods like martigales ad the Jaso ad Talagrad iequalities. The reader may cosult the survey paper of Specer [5] for further details. More recet applicatios ca be foud i [0] ad []. The mai objective of the curret paper is to provide a ew, stroger estimate o the probability defied above. This estimate is obtaied by combiig hypergraph argumets, somewhat similar to those used by Bollobás i [5], ad recet martigale results. We will prove that i a certai rage of the edge probability p(), the probability that G(, p) does ot cotai a idepedet set of size k 0, with k* k 0 c for some absolute costat c 0, is at most exp{( 2 /k 0 4 p)}. The exact formulatio of this result ad its proof are preseted i Sectio 2. Somewhat surprisigly, it turs out that the expoet i the estimate cited above is optimal up to a logarithmic factor. The proof of this is preseted i Sectio 3. The i Sectio 4 we demostrate how our ew boud ca be used to exted the scope of the results of [0] ad [] about the asymptotic value of the choice umber of radom graphs to smaller values of p(). Our argumet used to get a expoetial boud for the probability defied above ca i fact be viewed as a example of a geeral approach, for obtaiig expoetial bouds for probabilities of certai evets i product probability spaces. This geeral approach, discussed i Sectio 5, ca sometimes compete

3 ON THE PROBABILITY OF INDEPENDENT SETS IN RANDOM GRAPHS 3 successfully with the well kow Jaso iequality. Sectio 6, the fial sectio of the paper, is devoted to cocludig remarks. Throughout the paper we will use the stadard asymptotic otatio. I particular, a() b() meas a() o(b()), (a()) deotes a fuctio b() such that for some C 0, for sufficietly large b() Ca(), ad (a()) deotes a fuctio which is both O(a()) ad (a()). Also, f() g() meas lim 3 f()/g(). For the sake of clarity of presetatio we will systematically omit floor ad ceilig sigs at places where the choice of which is used does ot affect the argumet. 2. INDEPENDENT SETS IN RANDOM GRAPHS Let k 0 k 0 (, p) bedefied by k 0 maxk : k p k 2 4. () Oe ca show easily that k 0 satisfies k 0 2 log b (p) with b /( p). Also, it follows from kow results o the asymptotic value of the idepedece umber of G(, p) (see, e.g., [9]) that a.a.s. the differece betwee k 0 ad the idepedece umber of G(, p) is bouded by a absolute costat, as log as p() /2 for a positive 0. Theorem 2.. Let p() satisfy 2/5 l 6/5 p() for a absolute costat 0. The PrG, p k 0 e 2 /k 0 4 p. Proof. I case p is a costat, the result of the theorem follows easily from Jaso s iequality (see, e.g., [4], Chapter 0.3). Thus i the rest of the proof we will assume that p o(). Give a graph G o vertices ad a iteger k 0, a collectio of pairs of vertices of G is called a cover if every idepedet set of size k 0 i G cotais a pair from. We set X X(G) to be the miimum size of a cover i G. For the reader familiar with hypergraph termiology we ca defie X(G) as follows. Give G, defie a hypergraph H H(G) whose vertices are pairs of vertices of G ad whose edges are formed by takig all pairs of vertices i every idepedet set of G of size k 0. Thus H is a k ( 0 2 )-uiform hypergraph o ( 2 ) vertices, whose umber of edges is equal to the umber of idepedet sets of size k 0 i G. The a cover i G correspods to a vertex cover of the hypergraph H, ad X(G) is equal to the coverig umber of H. Whe G is distributed accordig to G(, p), the quatity X(G) becomes a radom variable. Our aim will be first to estimate from below the expectatio of X ad the to show that X is cocetrated. It may be oted that we use pairs of vertices i the defiitio of a cover, rather tha sigle vertices, i order to achieve a better cocetratio i Lemma 2.6 below, while larger sets of vertices would ot be suitable for Propositio 2.5. Lemma 2.2. E[X] 2 k 0 2.

4 4 KRIVELEVICH ET AL. Proof. Let Y be a radom variable coutig the umber of idepedet sets of size k 0 i G(, p). We deote by the expectatio of Y. The clearly EY k p 2 0 k0 4 by the defiitio of k 0. For a pair u, v V(G), let Z u,v be a radom variable coutig the umber of k 0 -subsets of V that cotai u ad v ad spa o edges except possibly the edge (u, v). [The edge (u, v) is permitted for ease of later aalysis.] Thus, if (u, v) E(G), the Z u,v is equal to the umber of idepedet sets of size k 0 that cotai both u ad v. If 0 E[Z u,v ], the 0 2 k 0 2 p k0 2. It is easy to see that, by defiitio, 0 / (k 0 2 / 2 ). Next, we set Z u,v Z u,v, Z u,v 2 0, 0, otherwise. We also defie Z u,v Z u,v. To fiish the proof of the lemma, we use three propositios. Propositio 2.3. For every graph G, X Y Z 2 0. Proof. Let be a optimal cover i G, X. Set 0 to be the set of pairs of vertices from coverig more tha 2 0 idepedet sets of size k 0, ad also set 0. Each pair u, v covers Z u,v idepedet sets of size k 0. Hece the set 0 covers at most {u,v}0 Z u,v Z such idepedet sets. The it follows that at least Y Z idepedet sets are covered by oly. As every pair i participates i at most 2 0 idepedet sets of size k 0, we get (Y Z )/(2 0 ). Therefore, X (Y Z )/(2 0 ), as required. Propositio 2.4. O k p 0 2 i 2 0. For each u, v V(G) ad all i with 2 0 i 2 k02, Pr[Z u,v i] Proof. Fix a pair u, v V(G) ad let U be the set of vertices i V{u, v} ot adjacet to either u or v. Bydefiitio the size of U is a biomially distributed radom variable with parameters 2 ad ( p) 2. Therefore by applyig stadard estimates for biomial distributios (see, e.g. [4, Theorems A. ad A.3]) to the size of V U we obtai that Pr U p 2 p l 2 e p/l4.

5 ON THE PROBABILITY OF INDEPENDENT SETS IN RANDOM GRAPHS 5 Deote by z the value of the radom variable Z u,v coditioal o the particular set U of size,( p) 2 p/l 2 ( p) 2 p/l 2. Let be the family of all subsets of U of size k 0 2. For every S, let Z S be the idicator radom variable takig value whe S spas o edges of G, ad value 0 otherwise. Clearly, z S Z S.Bydefiitio, the expected value ad the variace of z are equal to k 0 2 p k02 2, 2 VAR S Z S VARZ S COVZ S, Z S. S SS Clearly, if S, S have o commo pairs of vertices, the the evets Z S ad Z S are idepedet, implyig COV[Z S, Z S ] 0. Therefore, we eed to sum oly over those pairs S, S for which 2 S S k 0 3. This implies that 2 Ez k 0 2 i k 0 2 k 0 i 2 p2k i k 03 k 0 2 i2 p 2k02 k i2 k 0 2 i k 0 2 k 0 i 2 k 0 p 2 2. i Deote the i-th summad of the last sum by g(i), 2 i k 3. Oe ca check (see [4], Chap. 4.5 for a similar computatio) that the domiatig term is k 0 2 k 0 4 g2 2 k 0 2 k 0 2 p O k 0 4 p 2. Hece 2 O k 0 4 p 2 2. Next by applyig Chebyshev s iequality we obtai that Prz i Prz i 2 i 2. Usig the fact that ( p) 2 O(p/l 2 ) ad k 0 (l /p), we obtai 0 k 0 2 p k02 2 k 0 2 p k0 2 o p 2k04 k 02 o p k 0 3 k 0 3 o k 02 l 2 k 02 p 2 o. p 2k04

6 6 KRIVELEVICH ET AL. Now, to fiish the proof, ote that PrZ u,v i PrZ u,v i U p 2 p l 2 p l 2 2 PrU p 2 i 2 e p/l4 O k 0 4 p i 0 2. Here we used the estimate for 2 ad the facts that ( o()) ad that the maximal possible value of i 2 is ( k0 2) 2 e o(p/l4 ). Propositio 2.5. E[Z ] o(). Proof. We will use the followig easily prove statemet: If X is a iteger radom variable with fiitely may values, the, for every iteger s, iprx i sprx s PrX i. (2) is is For every pair (u, v) it ow follows from the defiitio of Z u,v ad Propositio 2.4 that EZ u,v iprz u,v i 2 0 PrZ u,v 2 0 i2 0 PrZ u,v i i O O k 0 4 p 0 2. k p i2 0 O k p 0 2 i 0 2 The we derive from the defiitio of Z ad the liearity of expectatio that E[Z ] u,v E[Z u,v ] O( 2 k 0 4 p 0 ) O( k 0 6 p ). Now applyig our assumptio o the edge 2 2 probability p(), we obtai the desired estimate. We ca ow complete the proof of Lemma 2.2. Recall that by Propositio 2.3, X (Y Z )/(2 0 ). Therefore, takig ito accout Propositio 2.5 ad the defiitios of ad 0, we derive EX EY EZ 2 0 o k 2, 0 as required. Lemma 2.6. For every 2 p t 0, Pr[X E[X] t] e t2 /2 2p. Proof. Notice that X is a edge Lipschitz radom variable, i.e., chagig a graph G i oe pair of vertices chages the value of X by at most. This is due to the fact that if a pair (u, v) becomes a o-edge, the i the worst case it ca be added to a optimal cover

7 ON THE PROBABILITY OF INDEPENDENT SETS IN RANDOM GRAPHS 7 to produce a ew cover. Whe applyig the edge exposure martigale to X, the maximal variace i the martigale is ( 2 ) p( p) 2 p/ 2. Therefore, the desired estimate of the lower tail of X follows from kow results o graph martigales (see, e.g., [4], Th ). We are ow i positio to fiish the proof of Theorem 2.. Clearly, a graph G cotais a idepedet set of size k 0 if ad oly if X 0. From Lemmas 2.2 ad 2.6, we obtai PrG k 0 PrX 0 PrX EX EX e (EX)2 /2 2p e 2 /k 0 4 p. 3. ON THE TIGHTNESS OF THEOREM 2. I this sectio we show that the expoet i the boud of Theorem 2. is tight up to logarithmic factors. Theorem 3.. (). The Let p() for a absolute costat 0. Defie k 0 k 0 (, p) by PrG, p k 0 e O2 l 2 /k 0 4 p. Proof. Set T 72 l 2, k 0 M 0 2p T. Our first goal will be to estimate from above the probability Pr[(G) k 0 E(G) M], where M M 0. Sice the distributio of G G(, p) coditioal o the evet E(G) M is idetical to the distributio of graph with M radom edges, we have PrG k 0 EG M 2 k 0 2 k 0 M M 2 0 k p T 2 2 k0 k p 0 k0 0 k M 2 2 k0 2 p p T 2 2 k0 2 k 0p k0 T 2 2 k0,

8 8 KRIVELEVICH ET AL. where we used the estimate ( ax b )( a b ) ( ab a )x i the secod iequality above. Returig to the defiitio () of k 0, we ca otice that k0 p 2 k0 6. Therefore, PrG k 0 EG M 6 e 2/ 2 Tk0. Substitutig the defiitio of T, we get Pr[(G) k 0 E(G) M] 6 7o() o(). As this estimate holds for every M M 0, it follows that PrG k 0 EG M 0 o. Also, due to the stadard estimates o the tails of a biomial radom variable, we have Pr[E(G) M 0 ] e T2 p 2. Combiig the two estimates above ad substitutig the value of T, we thus obtai PrG k 0 PrEG M 0 PrG k 0 EG M 0 oe T2 / 2p e 2 l 2 /k 4 p APPLICATIONS TO CHOOSABILITY OF RANDOM GRAPHS The choice umber ch(g) of a graph G is the miimum iteger k such that for every assigmet of a set S(v) ofk colors to every vertex v of G, there is a proper colorig of G that assigs to each vertex v a color from S(v). The choice umber was itroduced by Vizig [6] ad idepedetly by Erdős, Rubi, ad Taylor [8], ad the study of this parameter has received a cosiderable amout of attetio i recet years. I this sectio we cosider the asymptotic behavior of the choice umber of radom graphs. I their origial paper, Erdős, Rubi, ad Taylor [8] cojectured that almost surely ch(g(, /2)) o(). This was proved by Alo i []. Kah proved (see [2]) that almost surely chg, /2 og, /2 o/2 log 2. His result was exteded by Krivelevich [0], who determied the asymptotic value of ch(g(, p)) whe p() /4. At the same time Alo, Krivelevich, ad Sudakov [3] ad idepedetly Vu [7] showed that for all values of the edge probability p almost surely the choice umber of G(, p) has order of magitude (p/l(p)) (see also [2] for better costats). Here we combie Theorem 2. ad the ideas from [0] to prove the followig result. Theorem 4.. Let 0 /3 be a costat. If the edge probability p() satisfies /3 p() 3/4, the almost surely where b /( p). chg, p og, p o 2 log b p,

9 ON THE PROBABILITY OF INDEPENDENT SETS IN RANDOM GRAPHS 9 Sketch of the Proof. First ote that a.a.s. every subset of vertices of G(, p) of size at least m /l 4 cotais a idepedet set of size k 0 ( o())2 log b (p), where b /( p). Ideed, from Theorem 2., the fact that k 0 O(l /p) ad the assumptios o the value of p, it follows that the probability that there exists a set of m vertices that does ot spa a idepedet set of size k 0 is at most me m 2 k 4 0 p 2 e 3o o. Next we sketch how, give a typical graph G i G(, p) ad a family of lists S,..., S each of size /k 0 3p/l 2, we ca color G from these lists. Our colorig procedure cosists of two phases. As log as there exists a color c which appears i the lists of at least /l 4 of yet ucolored vertices, we do the followig. Deote by V 0 the set of those ucolored vertices whose color list cotais c. The V 0 /l 4. The V 0 spas a idepedet set I of size I k 0. We color all vertices of I by c, discard I, ad delete c from all lists. The total umber of deleted colors from each list S(v) durig the first phase caot exceed /k 0, as each time we remove a subset of size k 0. Let U deote the set of all vertices that are still ucolored after the first phase has bee completed. The lists of all vertices of U are still quite large, amely, S(u) 3p/l 2 for each u U. For a color c deote by W(c) the set of all vertices u U for which c is icluded i the correspodig list of colors S(u). We kow that W(c) /l 4 for each color c. Thus we expect that the degree of a vertex u i the spaed subgraph G[W(c)] is about O(p/l 4 ) S(u). If this ideed is the case for every color c ad every vertex u U, the each color c S(u) appears i the lists of oly few eighbors of u. The we ca color the vertices of U simply by pickig for each vertex a radom color from its list. Ufortuately the graph G[W(c)] ca have a few vertices of degree much higher tha O(p/l 4 ). We color those vertices first ad the treat the rest of U as idicated above. We omit techical details ad some additioal ideas required to complete the argumet, ad refer the reader to the paper of Krivelevich [0]. Next we cosider a differet model of radom graphs radom regular graphs. For a positive iteger-valued fuctio d d(), we defie the model G,d of radom regular graphs cosistig of all regular graphs o vertices of degree d with the uiform probability distributio. Our aim here is to provide the asymptotic value of the choice umber of G,d for d 4/5. As i the case of G(, p) we eed the followig lemma. Lemma 4.2. For every costat 0, if 4/5 d (3/4), the almost surely every subset of vertices of G,d of size at least m /l 4 cotais a idepedet set of size k 0 ( o())2 log b d, where b /( d). Proof. Let p d/. Wefirst eed a lower boud o the probability that a radom graph i G(, p) is regular. We use the result of Shamir ad Upfal [4, Eq. (35)] with () d, 2 for some 0, choosig w() () w(), to deduce that the umber of d-regular graphs o vertices is at least d/2 expod/22.

10 0 KRIVELEVICH ET AL. (Here there is a coditio o d(); growig faster tha log 2 is sufficiet.) It follows that for ay fixed 0 PrG, d/ is d-regular expd /2. O the other had, as we have already metioed i the proof of Theorem 4., the probability that the vertex set of G(, d/) cotais a subset of size m that does ot spa a idepedet set of size k 0 O( l d/d) is at most mexp m2 k 0 4 p 2 exp m2 k 4 0 p exp d3 o Comparig the last two expoets ad usig the assumptio d 4/5, we observe that the probability that G(, d/) is d-regular is much higher asymptotically tha the probability that G(, d/) cotais a large subset without a idepedet set of size k 0. Therefore, almost surely if d lies i the rage give i the assertio of the lemma, every subset of the vertices of G,d of size at least /l 4 spas a idepedet set of size k 0.. Usig this lemma, together with the ideas from [0] ad the upper boud o the size of idepedet set i G,d obtaied i [], oe ca deduce the followig theorem: Theorem 4.3. For every costat 0, if 4/5 d 3/4, the almost surely chg,d og,d o 2 log b d, where b /( d). Proof. The proof here is very similar to the proof of Theorem 4., ad we therefore restrict ourselves to just a few words about it, leavig techical details to the reader. To prove the lower boud for ch(g,d ) observe that obviously ch(g) (G) V(G)/(G) for every graph G. Pluggig i the estimate (G) (2 o())log b d for almost all graphs G i G,d, provided by Theorem 2.2 of [], we get the required lower boud. As for the upper boud, oe ca prove that almost surely the choice umber of G,d satisfies ch(g) /k 0 3d/l 2. The proof proceeds by essetially repeatig the proof of Theorem 4. for the edge probability p() d/. Give a d-regular graph G o vertices, satisfyig the coclusio of Lemma 4.2 ad havig some additioal properties, which hold almost surely i the probability space G,d, ad also give color lists {S(v) :v V(G)} of cardiality S(v) /k 0 3d/l 2, the colorig procedure starts by fidig idepedet sets of size k 0 i frequet colors (i.e., colors appearig i at least /l 4 lists). Oce such a set is foud i color c, we color all of its vertices by c, discard them, ad delete c from all lists. After this part of the colorig procedure has fiished, o color appears i more tha /l 4 vertices, ad each ucolored vertex still has a list of at least 3d/l 2 available colors. Moreover, for most ucolored vertices v V, most of the colors i the list S(v)

11 ON THE PROBABILITY OF INDEPENDENT SETS IN RANDOM GRAPHS appear i the lists of O(d/l 4 ) eighbors of v.wefirst treat few ucolored vertices which do ot have the above stated property, ad the color the rest by choosig colors at radom from correspodig lists. For more details the reader is referred to [0]. 5. A GENERAL SETTING The aim of this sectio is to show that the approach exhibited i the proof of Theorem 2. ca be applied i a much more geeral settig to obtai expoetial bouds for probabilities of certai evets. The bouds obtaied ca be better tha those provided by the celebrated Jaso iequality. Let H (V, E) be a hypergraph with V m vertices ad E k edges. We assume furthermore that H is r-uiform ad D-regular. Form a radom subset R V by Prv R p v, where these evets are mutually idepedet over v V. We wat to estimate the probability p 0 that the radom set R does ot cotai ay edge of H. Such a estimate is required frequetly i applicatios of the probabilistic method. The followig well-kow theorem, proved first by Jaso (see, e.g., [4], [9, Theorem 2.8]), usually gives a expoetial boud for p 0. To preset this theorem, let Y be the umber of edges of H spaed by R. We ca represet Y as I... I k, where I j are the idicator fuctios of the edges of H. Let E[Y], ad write I i I j if the correspodig edges itersect. Set i, j:ii I j E[I i I j ]. Theorem 5.. We have 2 p 0 exp. For ay vertex v of H let Y v deote the umber of edges f E(H) for which v f ad f{v} R; set v E[Y v ]. I may applicatios (especially those related to radom graphs) the probabilities p v all have the same value p. I this case, kp r ad v 0 Dp r for all v. Furthermore, it occurs frequetly that the sum i is domiated by the sum of those E[I i I j ], where the correspodig edges itersect i precisely oe vertex. I such a case, (r 0 ). Assumig 0, Jaso s iequality gives p 0 exp 2 exp r 0. (3) Our purpose here is to use the approach itroduced i Sectio 2 to show that uder a rather mild additioal assumptio (see Corollary 5.4), the followig holds: p 0 exp (4) pr 0. Iequality (4) is iterestig for two reasos. First, i certai applicatios p is very close to, ad therefore the term p i the deomiator yields a sigificat improvemet.

12 2 KRIVELEVICH ET AL. As we already saw i previous sectios, this is exactly the case for the probability of idepedet sets i radom graphs. For this problem a additioal term p is crucial, ad the boud give by iequality (4) is almost sharp. Secod, our proof is completely differet from that of Jaso (ad also from the alterative proof by Boppaa ad Specer [7]) ad the method might therefore be of idepedet iterest. From ow o we assume that p v p for all v V(H). Let X deote the coverig umber of the spaed subhypergraph H[R] (where the coverig umber of a hypergraph is the miimum umber of vertices eeded to cover all edges). Set mi(e[x], mp( p)). Theorem 5.2. The probability p 0 that the hypergraph H[R] has o edges satisfies p 0 exp 2 mp p. Proof. Similarly to Lemma 2.6, usig Theorem [4], we have that for every mp( p) t 0 [the maximum variace i the martigale is mp( p)] PrX EX t e t2 /4mpp. Clearly, a hypergraph H[R] cotais o edges if ad oly if X 0. Therefore, p 0 PrX 0 PrX EX e 2 /mpp. It is well kow that i a regular hypergraph, the coverig umber is at least the ratio betwee the umber of edges ad the degree. O the other had, the expectatio of the umber of edges of H[R] is, ad that of the degree of H[R] is 0. Thus, it is reasoable to thik that E[X] is (/ 0 ). The followig result shows that uder a additioal assumptio, this is ideed the case. Propositio 5.3. ( ). 0 Assume that VAR[Y v ] o(( 0 )/m) for all v. The E[X] Remark. We eed 0 istead of 0 i order to deal with the case 0. If 0, we ca replace 0 by 0. Notice also that / 0 kp mp D. r Now iequality (4) follows immediately from Theorem 5.2, Propositio 5.3, ad the above remark. We ote that the costat implicit i is idepedet of p, r, ad D. Corollary 5.4. for all v. The Assume that 0, /( 0 ) mp( p) ad VAR[Y v ] o( 0 /m) p 0 exp pr 0. We fiish this sectio with the sketch of the proof of Propositio 5.3.

13 ON THE PROBABILITY OF INDEPENDENT SETS IN RANDOM GRAPHS 3 Sketch of the Proof. Defie Z v Y v if Y v 2 0 ad Z v 0 otherwise. Similarly to the proof of Propositio 2.3, by settig Z v Z v, we obtai X Y Z 2 0. (5) Next, usig the assumptio that VAR[Y v ] o(( 0 )/m), Chebyshev s iequality ad applyig the same techiques as i the proof of Propositio 2.5, we ca show that VARY v / 0 o. EZ O v Now it follows immediately that EX EY EZ 2 0 o 2 0. This completes the proof. 6. CONCLUDING REMARKS Cosider the problem of estimatig the probability that a radom graph G(, p) has o cliques of cardiality t, with t fixed. I the settig of Sectio 5, defie a hypergraph H whose vertices are the edges of K ad whose edges are the t-cliques of G(, p). From Theorem 5., the probability that G(, p) has o t-cliques is at most exp( 2 /( )) where p t 2 t /t! ad ( 2 / 2 p). For fixed p with 0 [where 0 (/ 2 p)], the variace coditio i Corollary 5.4 is easy to verify. Hece, for all such p, Corollary 5.4 gives virtually the same result as Jaso s iequality, while its proof is etirely differet. The argumet also applies for graphs other tha cliques, but we do ot elaborate i this directio. A iterestig ad importat ope questio is to estimate the probability that G(, p) does ot cotai a idepedet set of size k ( )k 0, where k 0 is defied i (), ad is a small costat, or eve a fuctio of tedig to 0 very slowly as teds to ifiity. We cojecture the followig: Cojecture. PrG, p does ot cotai a idepedet set of size k exp 2 /k 2 p. This cojecture, if it holds, is best possible up to a logarithmic term i the expoet. It would immediately exted Theorem 4. to all p p() ( ) ad also give a short proof for Łuczak s result o the chromatic umber of G(, p) (by takig lim 3 () 0). Based o our method preseted i this paper, to prove the above cojecture, it suffices to show that the expectatio of the coverig umber of the correspodig hypergraph is

14 4 KRIVELEVICH ET AL. ( 2 /k) [istead of ( 2 /k 2 ) as show i the proof of Theorem 2.]. The followig speculatio might give the reader some ituitio why this could be the case. Cosider the complete hypergraph H com cosistig of all possible idepedet sets of size k. By Turá s theorem from extremal graph theory, the coverig umber of H com is ( 2 /k). As k is much smaller tha k 0, the expected umber of idepedet sets of size k i G(, p) is huge (roughly ( k ) ). So, the hypergraph correspodig to these idepedet sets looks typically like a fairly dese subhypergraph of H com ad oe may hope that such a hypergraph should have coverig umber close to that of H com, amely, ( 2 /k). REFERENCES [] N. Alo, Choice umbers of graphs; a probabilistic approach, Combiat Probab Comput (992), [2] N. Alo, Restricted colorigs of graphs, Surveys i combiatorics 993, Lodo Math. Soc. Lecture Notes Series 87, Ed. K. Walker, Cambridge, Cambridge Uiv. Press, 993, pp. 33. [3] N. Alo, M. Krivelevich, ad B. Sudakov, List colorig of radom ad pseudo-radom graphs, Combiatorica 9 (999), [4] N. Alo ad J. Specer, The probabilistic method, 2d editio, Wiley, New York, [5] B. Bollobás, The chromatic umber of radom graphs, Combiatorica 8 (988), [6] B. Bollobás ad P. Erdős, Cliques i radom graphs, Math Proc Camb Phil Soc 80 (976), [7] R. Boppaa ad J. Specer, A useful elemetary correlatio iequality, J Combiat Theory Ser A 50 (989), [8] P. Erdős, A. L. Rubi, ad H. Taylor, Choosability i graphs, Proc West Coast Cof Combiatorics, Graph Theory ad Computig, Cogressus Numeratium XXVI, 979, pp [9] S. Jaso, T. Łuczak, ad A. Ruciński, Radom graphs, Wiley, New York, [0] M. Krivelevich, The choice umber of dese radom graphs, Combiat Probab Comput 9 (2000), [] M. Krivelevich, B. Sudakov, V. H. Vu, ad N. C. Wormald, Radom regular graphs of high degree, Radom Struct Alg 8 (200), [2] M. Krivelevich ad V. H. Vu, Choosability i radom hypergraphs, J Combiat Theory Ser B 83 (200), [3] D. Matula, The largest clique size i a radom graph, Techical Report, Departmet of Computer Sciece, Souther Methodist Uiversity, Dallas, TX, 976. [4] E. Shamir ad E. Upfal, Large regular factors i radom graphs, Covexity ad graph theory (Jerusalem 98), North Hollad Math. Stud. 87, North Hollad, Amsterdam, 984, pp [5] J. Specer, Probabilistic methods i combiatorics, Proc It Cog Math Zürich, Birkhäuser, Basel, 994, pp [6] V. G. Vizig, Colorig the vertices of a graph i prescribed colors (i Russia), Diskret Aal 29, Met Diskret Aal Teor Kodov Shem 0 (976), 3 0. [7] V. H. Vu, O some degree coditios which guaratee the upper boud of chromatic (choice) umber of radom graphs, J Graph Theory 3 (999),