On the probability of independent sets in random graphs

Transcription

1 O the probability of idepedet sets i radom graphs Michael Krivelevich Bey Sudakov Va H Vu Nicholas C Wormald 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 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 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 1 Itroductio Let G, p deote as usual the probability space whose poits are graphs o labeled vertices {1,, }, 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 aas for short, if the probability that G, p satisfies A teds to 1 as the umber of vertices teds to ifiity Defie the followig quatity: k = max { k N : } 1 p k2 1 k Departmet of Mathematics, Raymod ad Beverly Sackler Faculty of Exact Scieces, Tel Aviv Uiversity, Tel Aviv 69978, Israel krivelev@posttauacil Research supported i part by a USA-Israel BSF Grat, by a grat from the Israel Sciece Foudatio ad by a Bergma Memorial Grat Departmet of Mathematics, Priceto Uiversity, Priceto, NJ 0850, USA ad Istitute for Advaced Study, Priceto, NJ 0850, USA bsudakov@mathpricetoedu Research supported i part by NSF grats DMS , CCR ad by the State of New Jersey Departmet of Mathematics, UCSD, La Jolla, CA vavu@ucsdedu, Web: vavu Research supported i part by Grat RB091G-Vu from UCSD, a Sloa fellowship ad a NSF Grat Departmet of Mathematics ad Statistics, Uiversity of Melboure VIC 3010, Australia ick@msuimelbeduau Web: ick Research supported by the Australia Research Coucil 1

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 1 It has bee kow for a log time [6], [13] that for large eough p = p say, for p ɛ for small eough costat ɛ > 0 aas 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 aas 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 [15] for further details More recet applicatios ca be foud i [10] ad [11] 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{ω 2 /k0 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 we demostrate how our ew boud ca be used to exted the scope of the results of [10] ad [11] 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 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 = ob, Ωa deotes a fuctio b such that for some C > 0, for sufficietly large b > Ca, ad Θa deotes a fuctio which is both Oa ad Ωa Also, f g meas lim f/g = 1 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

3 2 Idepedet sets i radom graphs Let k 0 = k 0, p be defied by k 0 = max { k : } 1 p k 2 1 k Oe ca show easily that k 0 satisfies k 0 2 log b p with b = 1/1 p Also, it follows from kow results o the asymptotic value of the idepedece umber of G, p see, eg, [9] that as the differece betwee k 0 ad the idepedece umber of G, p is bouded by a absolute costat, as log as p 1/2+ɛ for a positive ɛ > 0 Theorem 21 Let p satisfy 2/5 l 6/5 p 1 ɛ for a absolute costat 0 < ɛ < 1 The P r[αg, p < k 0 ] = e Ω 2 k 0 p Proof I case p is a costat, the result of the theorem follows easily from Jaso s iequality see, eg [], Chapter 103 Thus i the rest of the proof we will assume that p = o1 Give a graph G o vertices ad a iteger k 0, a collectio C of pairs of vertices of G is called a cover if every idepedet set of size k 0 i G cotais a pair from C We set X = XG to be the miimum size of a cover i G For the reader familiar with hypergraph termiology we ca defie XG as follows Give G, defie a hypergraph H = HG 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 XG is equal to the coverig umber of H Whe G is distributed accordig to G, p, the quatity XG 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 26 below, whilst larger sets of vertices would ot be suitable for Propositio 25 Lemma 22 E[X] = Ω 2 k 2 0 Proof Let Y be a radom variable coutig the umber of idepedet sets of size k 0 i G, p We deote by µ the expectatio of Y The clearly µ = E[Y ] = 1 p k 0 2 by the defiitio of k 0 k 0 3

4 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 EG, the Z u,v is equal to the umber of idepedet sets of size k 0 that cotai both u ad v If µ 0 = E[Z u,v ], the 2 µ 0 = 1 p k k 0 2 It is easy to see that, by defiitio, µ 0 /µ = Θk0 2/2 Next, we set { Z u,v + Z u,v, Z u,v > 2µ 0, = 0, otherwise We also defie Z + = u,v Z+ u,v To fiish the proof of the lemma, we use three propositios Propositio 23 For every graph G, X Y Z+ 2µ 0 Proof Let C be a optimal cover i G, C = X Set C 0 to be the set of pairs of vertices from C coverig more tha 2µ 0 idepedet sets of size k 0, ad also set C 1 = C \ C 0 Each pair u, v covers Z u,v idepedet sets of size k 0 Hece the set C 0 covers at most {u,v} C 0 Z u,v Z + such idepedet sets The it follows that at least Y Z + idepedet sets are covered by C 1 oly As every pair i C 1 participates i at most 2µ 0 idepedet sets of size k 0, we get C 1 Y Z + /2µ 0 Therefore X C 1 Y Z + /2µ 0, as required Propositio 2 For each u, v V G ad all i with 2µ 0 k O 0 pµ i µ 0 2 i 2 k 0 2, P r[zu,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 By defiitio the size of U is a biomially distributed radom variable with parameters 2 ad 1 p 2 Therefore by applyig stadard estimates for biomial distributios see, eg [, Theorems A11 ad A13] to the size of V U we obtai that [ U P r 1 p 2 > p ] l 2 < e Ω p l Deote by z 1 the value of the radom variable Z u,v coditioal o the particular set U of size 1, 1 p 2 p/ l p 2 + p/ l 2 Let S be the family of all subsets of U of size k 0 2 For every S S let Z S be the idicator radom variable takig value 1 whe S spas o edges of G, ad value 0 otherwise Clearly, z 1 = S S Z S By defiitio, the expected value ad the variace of z 1 are equal to µ 1 = 1 1 p k 0 2 2, k 0 2

5 σ 2 1 = V AR[ S S Z S ] = S S V AR[Z S ] + S S S COV [Z S, Z S ] Clearly, if S, S S have o commo pairs of vertices, the the evets Z S = 1 ad Z S = 1 are idepedet, implyig COV [Z S, Z S ] = 0 Therefore we eed to sum oly over those pairs S, S S for which 2 S S k 0 3 This implies: σ1 2 1 E[z 1 ] + = µ 1 + µ 2 1 k 0 2 k0 3 i=2 k 0 3 k0 2 1 k 0 +2 i k 0 i 2 1 i=2 k 0 2 k0 2 1 k i k 0 i p i 2 1 [ 1 p 2k i 2 1 p 2 k 0 2 Deote the i-th summad of the last sum by gi, 2 i k 3 Oe ca check see [], Chapter 5 for a similar computatio that the domiatig term is g2 = k0 2 1 k k 0 1 k p 1 = O k 0 p Hece σ1 2 = O k 0 p µ Next by applyig Chebyshev s iequality we obtai that 1 P r[z 1 i] P r[ z µ 1 i µ 1 ] 2 1 σ 2 1 i µ 1 2 Usig the fact that 1 = 1 p 2 + Θp/ l 2 ad k 0 = Θl /p we obtai µ 1 µ 0 = 1 k 0 2 k p k p = 1 1 k k k k0 2 1 = 1 + o1 1 p p 2k 0 = 1 + o1 1 p 2k 0 p k0 2 = 1 + o1 1 + Θ l 2 = 1 + o1 2 ] 1 k0 2 Now to fiish the proof ote that [ ] [ ] p U ] P r Z u,v i P r Z u,v i U = 1 p 2 + Θ l 2 + P r[ 1 p 2 p > Θ l 2 = O σ 2 1 p l i µ e Ω k 0 pµ i µ 0 2 Here we used the estimate for σ 2 1 ad the facts that 1 = 1 + o1 ad that the maximal possible value of i 2 is k = e op/l 5

6 Propositio 25 E[Z + ] = oµ 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, ip r[x = i] = sp r[x > s] + P r[x i] 2 i>s i>s For every pair u, v it ow follows from the defiitio of Z u,v ad Propositio 2 that E[Z + u,v] = ip r[z u,v = i] = 2µ 0 P r[z u,v > 2µ 0 ] + i>2µ 0 k 0 pµ µ 0 µ = 2µ 0 O = O k 0 pµ 0 2 k O 0 pµ 2 0 i>2µ 0 i>2µ 0 P r[z u,v i] 2 1 i µ 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 k0 pµ 0 k 6 = O 0 p µ Now applyig our assumptio o the edge probability p, we obtai the desired 2 2 estimate We ca ow complete the proof of Lemma 22 Recall that by Propositio 23, X Y Z + /2µ 0 Therefore, takig ito accout Propositio 25 ad the defiitios of µ ad µ 0, we derive: as required E[X] E[Y ] E[Z+ ] 1 o1µ 2 = = Ω, 2µ 0 2µ 0 Lemma 26 For every 2 p > t > 0, P r[x E[X] t] e t p Proof Notice that X is a edge Lipschitz radom variable, ie chagig a graph G i oe pair of vertices chages the value of X by at most oe 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 to produce a ew cover Whe applyig the edge exposure martigale to X, the maximal variace i the martigale is 2 p1 p 2 p/2 Therefore, the desired estimate of the lower tail of X follows from kow results o graph martigales see, eg [], Th 73 We are ow i positio to fiish the proof of Theorem 21 Clearly, a graph G cotais a idepedet set of size k 0 if ad oly if X > 0 From Lemmas 22 ad 26 we obtai: [ ] P r[αg < k 0 ] = P r[x = 0] = P r X E[X] E[X] e E[X]2 2 2 p k 2 0 = e Ω 2 k 0 p 6

7 3 O the tightess of Theorem 21 I this sectio we show that the expoet i the boud of Theorem 21 is tight up to logarithmic factors Theorem 31 Let p 1 ɛ for a absolute costat ɛ > 0 Defie k 0 = k 0, p by 1 The Proof Set P r[αg, p < k 0 ] = e O 2 l 2 k 0 p T = 72 l k0 2, M 0 = p + T 2 Our first goal will be to estimate from above the probability P r[αg k 0 EG = M], where M M 0 Sice the distributio of G = G, p coditioal o the evet EG = M is idetical to the distributio of graph with M radom edges, we have P r[αg k 0 EG = M] where we used the estimate a x a b b = 1 a b a 2 k 0 2 k 0 M 2 M k 0 1 p T 2 k 0 1 M 2 k 02 k 02 1 p k p T k p k 0 k 0 k 02 1 p ko 2 1 T, 2 defiitio 1 of k 0, we ca otice that k 0 1 p k Therefore: x i the secod iequality above Returig to the T k 0 P r[αg k 0 EG = M] 6 e 2 2 Substitutig the defiitio of T, we get P r[αg k 0 EG = M] = 6 7+o1 = o1 As this estimate holds for every M M 0, it follows that P r[αg < k 0 EG M 0 ] = 1 o1 7

8 Also, due to the stadard estimates o the tails of a biomial radom variable we have P r[ EG M 0 ] = e Θ T obtai: 2 2 p Combiig the two estimates above ad substitutig the value of T, we thus P r[αg < k 0 ] P r[ EG M 0 ]P r[αg < k 0 EG M 0 ] 1 o1e Θ T 2 2 p = e Θ 2 l 2 k 0 p Applicatios to choosability of radom graphs The choice umber chg of a graph G is the miimum iteger k such that for every assigmet of a set Sv of k colors to every vertex v of G, there is a proper colorig of G that assigs to each vertex v a color from Sv The choice umber was itroduced by Vizig [16] ad idepedetly by Erdős, Rubi ad Taylor [8] ad the study of this parameter 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 chg, 1/2 = o This was proved by Alo i [1] Kah proved see [2] that almost surely chg, 1/2 = 1 + o1χg, 1/2 = 1 + o1/2 log 2 His result was exteded by Krivelevich [10], who determied the asymptotic value of chg, p whe p 1/ At the same time Alo, Krivelevich ad Sudakov [3] ad idepedetly Vu [17] showed that for all values of the edge probability p almost surely the choice umber of G, p has order of magitude Θp/ lp see also [12] for better costats Here we combie Theorem 21 ad the ideas from [10] to prove the followig result Theorem 1 Let 0 < ɛ < 1/3 be a costat p 3/ the almost surely where b = 1/1 p If the edge probability p satisfies 1/3+ɛ chg, p = 1 + o1χg, p = 1 + o1 2 log b p, Sketch of the proof First ote that aas every subset of vertices of G, p of size at least m / l cotais a idepedet set of size k 0 = 1 o12 log b p, where b = 1/1 p Ideed, from Theorem 21, the fact that k 0 = Ol /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 e Ω m 2 k 0 p 2 e 1+3ɛ o1 = o1 m 8

9 Next we sketch how, give a typical graph G i G, p ad a family of lists S 1,, 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 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 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 Sv 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, Su 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 Su We kow that W c / l for each color c Thus we expect that the degree of a vertex u i the spaed subgraph G[W c] is about Op/ l Su If this ideed is the case for every color c ad every vertex u U, the each color c Su 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 Op/ l 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 [10] 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 /5 As i the case of G, p we eed the followig lemma Lemma 2 For every costat ɛ > 0, if /5+ɛ d 3/, the almost surely every subset of vertices of G,d of size at least m = / l cotais a idepedet set of size k 0 = 1 o12 log b d, where b = / d Proof Let p = d/ We first eed a lower boud o the probability that a radom graph i G, p is regular We use the result of Shamir ad Upfal [1, equatio 35] with φ = d, θ = δ for some δ > 0, choosig w φ = w 1 δ, to deduce that the umber of d-regular graphs o vertices is at least 2 exp Od 1/2+2δ d/2 Here there is a coditio o d; growig faster tha log 2 is sufficiet It follows that for ay fixed δ > 0 P r[g, d/ is d-regular] exp d 1/2+δ 9

10 O the other had as we have already metioed i the proof of Theorem 1, 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 exp m m 2 Ω k0 p 2 exp m 2 Ω k0 p = exp d 3 1 o1 Comparig the last two expoets ad usig the assumptio d /5+ɛ, 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 spas a idepedet set of size k 0 Usig this lemma, together with the ideas from [10] ad the upper boud o the size of idepedet set i G,d obtaied i [11], oe ca deduce the followig theorem: Theorem 3 For every costat ɛ > 0, if /5+ɛ d 3/, the almost surely where b = / d chg,d = 1 + o1χg,d = 1 + o1 2 log b d, Proof The proof here is very similar to the proof of Theorem 1, ad we therefore restrict ourselves to just a few words about it, leavig techical details to the reader To prove the lower boud for chg,d observe that obviously chg χg V G /αg for every graph G Pluggig i the estimate αg = 2 + o1 log b d for almost all graphs G i G,d, provided by Theorem 22 of [11], we get the required lower boud As for the upper boud, oe ca prove that almost surely the choice umber of G,d satisfies chg /k 0 + 3d/ l 2 The proof proceeds by essetially repeatig the proof of Theorem 1 for the edge probability p = d/ Give a d-regular graph G o vertices, satisfyig the coclusio of Lemma 2 ad havig some additioal properties, which hold almost surely i the probability space G,d, ad also give color lists {Sv : v V G} of cardiality Sv = /k 0 + 3d/ l 2, the colorig procedure starts by fidig idepedet sets of size k 0 i frequet colors ie colors appearig i at least / l 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 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 Sv appear i the lists of Od/ l eighbors of v We first 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 [10] 10

11 5 A geeral settig The aim of this sectio is to show that the approach exhibited i the proof of Theorem 21 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 furthermore that H is r-uiform ad D-regular Form a radom subset R V by P r[v R] = p v, where these evets are mutually idepedet over v V We assume 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, eg, [], [9, Theorem 218], 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 µ = E[Y ], ad write I i I j if the correspodig edges itersect Set = i,j: I i I j E[I i I j ] Theorem 51 We have p 0 exp µ2 µ + For ay vertex v of H let Y v deote the umber of edges f EH for which v f ad f \ {v} R; set µ 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, µ = kp r ad µ v = µ 0 = Dp r 1 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, = Θrµµ 0 Assumig µ 0 1, Jaso s iequality gives µ 2 µ p 0 exp Θ = exp Ω 3 µ + rµ 0 Our purpose here is to use the approach itroduced i Sectio 2 to show that uder a rather mild additioal assumptio see Corollary 5, the followig holds: µ p 0 exp Ω 1 prµ 0 Iequality is iterestig for two reasos First, i certai applicatios p is very close to 1 ad therefore the term 1 p i the deomiator yields a sigificat improvemet 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 1 p is crucial, ad the boud give by iequality 11

12 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 τ = mi E[X], mp1 p Theorem 52 The probability p 0 that the hypergraph H[R] has o edges satisfies: τ 2 p 0 exp Ω mp1 p Proof Similarly to Lemma 26, usig Theorem 73 [], we have that for every mp1 p > t > 0 the maximum variace i the martigale is mp1 p, P r[x E[X] t] e t 2 mp1 p Clearly, a hypergraph H[R] cotais o edges if ad oly if X = 0 Therefore p 0 = P r[x = 0] P r[x EX τ] e Ω τ 2 mp1 p 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 µ ad that of the degree of H[R] is µ 0 Thus, it is reasoable to thik that E[X] is Ωµ/µ 0 The followig result shows that uder a additioal assumptio, this is ideed the case Propositio 53 Assume that V AR[Y v ] = o µµ 0 + 1/m for all v The E[X] = Ω µ µ 0 +1 Remark We eed µ istead of µ 0 i order to deal with the case µ 0 1 If µ 0 1, we ca replace µ by µ 0 Notice also that µ/µ 0 = kp D = mp r Now iequality follows immediately from Theorem 52, Propositio 53 ad the above remark We ote that the costat implicit i Ω is idepedet of p, r ad D Corollary 5 Assume that µ 0 1, µ/µ mp1 p ad V AR[Y v ] = oµµ 0 /m for all v The µ p 0 exp Ω 1 prµ 0 We fiish this sectio with the sketch of the proof of Propositio 53 12

13 Sketch of the proof Defie Z v = Y v if Y v 2µ ad Z v = 0 otherwise Similarly to the proof of Propositio 23, by settig Z + = v Z v, we obtai X Y Z+ 2µ Next, usig the assumptio that V AR[Y v ] = o µµ 0 +1/m, Chebyshev s iequality ad applyig the same techiques as i the proof of Propositio 25, we ca show that E[Z + ] = O v V AR[Y v ]/µ 0 = oµ Now it follows immediately that This completes the proof τ = E[X] E[Y ] E[Z+ ] 2µ µ = 1 + o1 2µ Cocludig 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 51, the probability that G, p has o t-cliques is at most exp µ 2 /µ+ where µ p t 2 t /t! ad = Θµ 2 / 2 p For fixed p with µ 0 > 1 where µ 0 = Θµ/ 2 p, the variace coditio i Corollary 5 is easy to verify Hece, for all such p, Corollary 5 gives virtually the same result as Jaso s iequality, whilst 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 = 1 ɛk 0, where k 0 is defied i 1, ad ɛ is a small costat, or eve a fuctio of tedig to 0 very slowly as teds to ifiity We cojecture that Cojecture [ ] P r G, p does ot cotai a idepedet set of size k exp Ω 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 1 to all p = p = Ω 1+ɛ ad also give a short proof for Luczak s result o the chromatic umber of G, p by takig lim ɛ = 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 Ω 2 /k istead of Ω 2 /k 2 as show i the proof of Theorem 21 The followig speculatio might give the reader 13

14 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 Ω 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 ɛ So, the hypergraph correspodig to these idepedet sets looks typically like a fairly dese sub-hypergraph of H com ad oe may hope that such a hypergraph should have coverig umber close to that of H com, amely Ω 2 /k Refereces [1] N Alo, Choice umbers of graphs; a probabilistic approach, Combiatorics, Probability ad Computig , [2] N Alo, Restricted colorigs of graphs, i Surveys i Combiatorics 1993, Lodo Math Soc Lecture Notes Series 187 K Walker, ed, Cambridge Uiv Press 1993, 1 33 [3] N Alo, M Krivelevich ad B Sudakov, List colorig of radom ad pseudo-radom graphs, Combiatorica , [] N Alo ad J Specer, The probabilistic method, 2 d ed, Wiley, New York, 2000 [5] B Bollobás, The chromatic umber of radom graphs, Combiatorica , 9 55 [6] B Bollobás ad P Erdős, Cliques i radom graphs, Math Proc Camb Phil Soc , [7] R Boppaa ad J Specer, A useful elemetary correlatio iequality, J Comb Th Ser A , [8] P Erdős, A L Rubi ad H Taylor, Choosability i graphs, Proc West Coast Cof o Combiatorics, Graph Theory ad Computig, Cogressus Numeratium XXVI, 1979, [9] S Jaso, T Luczak ad A Ruciński, Radom graphs, Wiley, New York, 2000 [10] M Krivelevich, The choice umber of dese radom graphs, Combiatorics, Probability ad Computig , [11] M Krivelevich, B Sudakov, V H Vu ad NC Wormald, Radom regular graphs of high degree, Radom Structures ad Algorithms , [12] M Krivelevich ad V H Vu, Choosability i radom hypergraphs, J Comb Theory Ser B ,

15 [13] D Matula, The largest clique size i a radom graph, Tech Rep Dept Comp Sci, Souther Methodist Uiversity, Dallas, Texas, 1976 [1] E Shamir ad E Upfal, Large regular factors i radom graphs, Covexity ad graph theory Jerusalem 1981, North Hollad Math Stud 87, North Hollad, Amsterdam-New York 198, pp [15] J Specer, Probabilistic methods i combiatorics, i Proceedigs It Cogress of Math Zürich, 199, Birkhäuser,Basel, [16] V G Vizig, Colorig the vertices of a graph i prescribed colors i Russia, Diskret Aaliz No 29, Metody Diskret Aal v Teorii Kodov i Shem , 3-10 [17] VH Vu, O some degree coditios which guaratee the upper boud of chromatic choice umber of radom graphs, Joural of Graph Theory ,