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The coupling method for inhomogeneous random intersection graphs. 3 Katarzyna Rybarczyk 1 0 Faculty of Mathematics and Computer Science, 2 n Adam Mickiewicz University, 60–769 Poznan´, Poland a [email protected] J 3 1 Abstract ] O We present new results concerning threshold functions for a wide family of ran- C dom intersection graphs. To this end we apply the coupling method used for estab- . h lishing threshold functions for homogeneous random intersection graphs introduced t by Karon´ski, Scheinerman, and Singer–Cohen. In the case of inhomogeneous ran- a m dom intersection graphs the method has to be considerably modified and extended. [ By means of the altered method we are able to establish threshold functions for a general random intersection graph for such properties as k-connectivity, match- 3 v ing containment or hamiltonicity. Moreover using the new approach we manage to 6 sharpenthebestknownresultsconcerninghomogeneousrandomintersection graph. 6 4 0 . 1 Introduction 1 0 3 The first random intersection graph model was introduced by Karon´ski, Scheinerman, 1 : and Singer–Cohen [11]. Since than it has been attracting attention mainly because of v i its wide applications, for example: ”gate matrix layout” for VLSI design (see e.g. [11]), X cluster analysis and classification (see e.g. [9]), analysis of complex networks (see e.g. r a [7, ?]), secure wireless networks (see e.g. [2]) and epidemics ([6]). Several generalisations of the model has been proposed, mainly in order to adapt it to use in some particular purpose. In this paper we consider the (n,m,p) model studied for example in [1, 5, 12]. G Alternative ways of generalizing the model defined in [11] are given for example in [7] and [9]. In a random intersection graph (n,m,p) there is a set of n vertices = v ,...,v , 1 n G V { } an auxiliary set of m = m(n) features = w ,...,w , and a vector p(n) = 1 m(n) W { } (p ,...,p ) such that p (0;1), for each 1 i m. Each vertex v adds 1 m(n) i ∈ ≤ ≤ ∈ V a feature w to its feature set W(v) with probability p independently of all other i i ∈ W properties and features. Any two vertices v,v are connected by an edge in (n,m,p) ′ ∈ V G if W(v) and W(v ) intersect. If p(n) = (p,...,p) for some p (0;1) then (n,m,p) is a ′ ∈ G random intersection graph defined in [11]. We denote it by (n,m,p). G The random intersection graph model is very flexible and its properties change a lot if we alter the parameters. For example (n,m,p) for some ranges of parameters behaves G 1 similarly to a random graph with independent edges (see [8, 13]) but in some cases it exhibit large dependencies between edge appearance (see for example [11, 15]). It was proved in [14] that in both cases (n,m,p) may be coupled with a random graph with G independent edges so that with probability tending to 1 as n , (n,m,p) is an → ∞ G overgraph of a graph with independent edges. It is also explained how this coupling may be used to obtain sharp results on threshold functions for (n,m,p). Such properties G as connectivity, a Hamilton cycle containment or a matching containment are given as examples. Ingeneral, thecouplingtechniqueprovidesaveryelegantmethodtogetbounds on threshold functions for random intersection graphs for a large class of properties. However theproofpresentedin[14]cannotbestraightforwardgeneralisedto (n,m,p) G with arbitrary p(n). First of all it differentiates between cases np 0 and np . → → ∞ Moreover the method does not give sharp results for np tending to a constant. In this article we modify and extend the techniques used in [14] in order to overcome these constraints. First of all, to get the general result, we couple (n,m,p) with an auxiliary G random graph which does not have fully independent edges. Therefore we need to prove some additional facts about the auxiliary random graph. Moreover we need sharp bounds on the minimum degree threshold function for (n,m,p). Due to edge dependencies, G estimation of moments of the random variable counting vertices with a given degree in (n,m,p) is complicated. Therefore we suggest a different approach to resolve the G problem. We divide (n,m,p) into subgraphs so that the solution of a coupon collector G problemcombinedwiththemethodofmomentsprovidetheanswer. Thisnewapproachto thecoupling methodallowustoobtainbetter resultsonthresholdfunctions for (n,m,p) G and by this means resolve open problems left over in [14]. Concluding, we provide a general method to establish bounds on threshold functions for many properties for (n,m,p). By means of the method we are able to obtain sharp G thresholds for k–connectivity, perfect matching containment and hamiltonicity for the general model. Last but not least we considerably improve known results concerning (n,m,p). G All limits in the paper are taken as n . Throughout the paper we use standard → ∞ asymptotic notation o( ), O( ), Ω( ), Θ( ), , , and defined as in [10]. By Bin(n,p) · · · · ∼ ≪ ≫ and Po(λ) we denote the binomial distribution with parameters n, p and the Poisson distribution with expected value λ, respectively. We also use the phrase “with high probability” to say with probability tending to one as n tends to infinity. All inequalities hold for n large enough. 2 Main Results In the article we compare random intersection graph (n,m,p) with a sum of a random G graph with independent edges G (n,pˆ ) and a random graph G (n,pˆ ) constructed on the 2 2 3 3 base of a random 3–uniform hypergraph with independent hyperedges. Generally, for any pˆ= pˆ(n) [0;1] and i = 2,...,n, let H (n,pˆ) be an i–uniform hypergraph with the vertex i ∈ set in which each i–element subset of is added to the hyperedge set independently V V with probability pˆ. G (n,pˆ) is a graph with the vertex set and an edge set consisting i V of those two element subsets of which are subsets of at least one hyperedge of H (n,pˆ). i V 2 We consider monotone graph properties of random graphs. For the family of all G graphs with the vertex set , we call a property if it is closed under isomorphism. V A ⊆ G Moreover is increasing if G implies G for all G such that E(G) E(G) ′ ′ ′ A ∈ A ∈ A ∈ G ⊆ anddecreasing if isincreasing. Increasingpropertiesareforexample: k–connectivity, G\A containing a perfect matching and containing a Hamilton cycle. Let p = (p ,...,p ) be such that p (0,1), for all 1 i m. Define 1 m i ∈ ≤ ≤ m S = np 1 (1 p )n 1 ; 1 i i − − − i=1 X (cid:0) (cid:1) m 1 (1 2p )n i S = np 1 − − ; 2 i − 2np i=1 (cid:18) i (cid:19) (1) X m 1 (1 2p )n S = np − − i (1 p )n 1 ; 3 i i − 2np − − i=1 (cid:18) i (cid:19) X m n S = t pt(1 p )n t, for t = 2,3,...,n. 1,t t i − i − i=1 (cid:18) (cid:19) X The following theorem is an extension of the result obtained in [14]. Theorem 1. Let S , S and S be given by (1). For some function ω tending to infinity 1 2 3 let S ω√S 2S2n 2 pˆ= 2 − 2 − 2 − ; 2 n 2 S1−3S3−ω2(cid:0)√(nS(cid:1))1−2S12n−2, for S3 ≫ √S1 and ω2 ≪ S3/√S1; pˆ = 2 (2) 2 S1−ω√2S(1n−)2S12n−2, for S3 = O(√S1); 2 pˆ = S3−ω√S(1n−)6S32n−3, for S3 ≫ √S1 and ω2 ≪ S3/√S1; 3 3  0, for S = O(√S ).  3 1 If S and S = o(n2) then for any increasing property . 1 1 → ∞ A (3) liminfPr G (n,pˆ) limsupPr (n,m,p) , 2 n { ∈ A} ≤ n {G ∈ A} →∞ →∞ (4) liminfPr G (n,pˆ ) G (n,pˆ ) limsupPr (n,m,p) . 2 2 3 3 n { ∪ ∈ A} ≤ n {G ∈ A} →∞ →∞ Remark 1. Assumption S is natural since for S = o(1) with high probability 1 1 → ∞ (n,m,p) is an empty graph. G Remark 2. S is the expected number of edges in G (n,pˆ ). If S = O(√S ) then by 3 3 3 3 1 Markov’s inequality with high probability the number of edges in G (n,pˆ ) is at most 3 3 ω√S . Thus S = S S = S +O(ω√S ) and the bound provided by (3) is as good as 1 2 1 3 1 1 − the one taking into consideration the edges from G (n,pˆ ). 3 3 3 Remark 3. Theorem is also valid for S = Ω(n2) but with 1 S ω√S 2 2 pˆ= 1 exp − ; − − 2 n ! 2 1 exp S1 (cid:0)3S(cid:1)3 ω√S1 , for S √S and ω S /√S ; − − − 2(n−) 3 ≫ 1 ≪ 3 1 pˆ =  (cid:18) 2 (cid:19) 2 1−exp −S1−2(ωn√)S1 , for S3 = O(√S1); (cid:18) 2 (cid:19)  1 exp S3 ω√S1 , for S √S and ω S /√S ; pˆ = − − −(n) 3 ≫ 1 ≪ 3 1 3  (cid:18) 3 (cid:19) 0, for S = O(√S ).  3 1 Denote by k, and the following graph properties: a graph is k–connected, C PM HC has a perfect matching and has a Hamilton cycle, respectively. We will use Theorem 1 to establish threshold functions for , and in (n,m,p). By we denote k k C PM HC G C here vertex connectivity. From the proof it follows that the threshold function for edge connectivity is the same as this for . k C For any sequence c with limit we write n 0 for c ; n → −∞ (5) f(c ) = e e−c for c c ( ; ); n  − n → ∈ −∞ ∞ 1 for c . n → ∞ Theorem 2. Let max1 i mpi =o((lnn)−1) and S1 and S1,2 be given by (1). ≤≤ (i) If S = n(lnn+c ), then 1 n lim Pr (n,m,p) = f(c ), 1 n n {G ∈ C } →∞ where f(c ) is given by (5). n (ii) Let k be a positive integer and a = S1,2. If S = n(lnn+(k 1)lnlnn+c ), then n S1 1 − n 0 for c and a a (0;1]; n n lim Pr (n,m,p) = → −∞ → ∈ k n→∞ {G ∈ C } (1 for cn → ∞. Assumption max p = o((lnn) 1) is necessary to avoid awkward cases. The 1 i m i − ≤≤ problem is explained in more detail in Section 4. A straightforward corollary of the above theorem is that for S = n(lnn+c ), c and any k = 1,2,...,n. 1 n n → −∞ lim Pr (n,m,p) = 0 and lim Pr (n,m,p) = 0. k n {G ∈ C } n {G ∈ HC} →∞ →∞ Theorem 3. Let max p = o((lnn) 1) and S be given by (1). If S = n(lnn+c ) 1 i m i − 1 1 n ≤≤ then lim Pr (2n,m,p(2n)) = f(c ), 2n n {G ∈ PM} →∞ where f( ) is given by (5). · 4 Theorem 4. Let max p = o((lnn) 1), S and S be given by (1) and a = S1,2. 1≤i≤m i − 1 1,2 n S1 If S = n(lnn+lnlnn+c ), then 1 n 0 for c and a a (0;1]; n n lim Pr (n,m,p) = → −∞ → ∈ n→∞ {G ∈ HC} (1 for cn → ∞. Already simple corollaries of Theorems 2–4 give sharp threshold functions for (n,m,p). G For example. Corollary 1. Let m ln2n and p(1 (1 p)n 1) = (lnn+c )/m. Then − n ≫ − − lim Pr (n,m,p) = f(c ) 1 n n {G ∈ C } →∞ and lim Pr (2n,m,p) = f(c ), 2n n {G ∈ PM} →∞ where f( ) is given by (5). · In particular we may state the following extension of the result from [17]. Corollary 2. Let b be a sequence, β and γ be constants such that βγ(1 e γ) = 1. If n − − γ b n m = βnlnn and p = 1+ then n lnn (cid:18) (cid:19) e γγ − lim Pr (n,m,p) = f 1+ b n {G ∈ C1} 1 e γ n →∞ (cid:18)(cid:18) − − (cid:19) (cid:19) and e γγ − lim Pr (2n,m,p(2n)) = f 1+ b , n {G ∈ PM} 1 e γ 2n →∞ (cid:18)(cid:18) − − (cid:19) (cid:19) where f( ) is given by (5). · Sometimes the method of the proof enable to improve the best known results concerning (n,m,p) even more. G Theorem 5. Let m ln2n and ≫ lnn+ln max 1,ln npe−nplnn +c 1 e−np n (6) p(1 (1 p)n 1) = − . − − − (cid:16) n m(cid:16) (cid:17)o(cid:17) Then 0 for c ; n lim Pr (n,m,p) = → −∞ n→∞ {G ∈ HC} (1 for cn → ∞. 5 Theorem 6. Let m ln2n and k be a positive integer. If ≫ k 1 lnn+ln max 1,(np)k 1 e−nplnn − + e−nplnn +c − 1 e−np 1 e−np n p(1 (1 p)n 1) = (cid:18) (cid:26) (cid:18) − − (cid:19)(cid:27)(cid:19) , − (cid:16) (cid:17) − − m then 0 for c ; n lim Pr (n,m,p) = → −∞ k n→∞ {G ∈ C } (1 for cn → ∞. One of the question posed in [14] concerned the range of m = m(n) for which the thresholdfunctionfor for (n,m,p)coincideswiththisforδ( (n,m,p)) 1. Moreover k C G G ≥ we may ask when threshold function for for (n,m,p) is the same as this for for k k C G C G (n,pˆ) with pˆ= mp2. Theorem 6 gives a final answer to these questions. 2 Corollary 3. Let k be a positive integer. If lnn+cn, for ln2n m nlnn p(1 (1 p)n 1) = m ≪ ≪ lnlnn − − − (lnn+(k−1)lnlnn+cn, for m = Ω(nlnn); m then 0 for c ; n lim Pr (n,m,p) = → −∞ k n→∞ {G ∈ C } (1 for cn → ∞. The proof is divided as follows. Section 3 describes the coupling used to establish threshold functions and presents the proof of Theorem 1. In Section 4 we give minimum degree thresholds for (n,m,p). Section 5 is dedicated to the properties of the auxiliary G randomgraphsusedinthecouplingestablishedinTheorem1. Theproofsoftheremaining theorems are presented in Section 6. 3 Coupling In this section we present a proof of Theorem 1. In the proof we use auxiliary random graph models G (n,M), i = 2,3,...,n, in which M is a random variable with non– i negative integer∗values. For i = 2,...,n, G (n,M) is constructed on the basis of a i random hypergraph H (n,M). H (n,M) is a∗random hypergraph with the vertex set i i ∗ ∗ V in which the hyperedge set is constructed by sampling M times with repetition elements fromtheset of all i–element subsets of (all sets which arechosen several times areadded V only once to the hyperedge set). G (n,M) is a graph with the vertex set in which i ∗ V v,v are connected by an edge if v,v is contained in at least one of the hyperedges ′ ′ ∈ V { } of H (n,M). If M equals a constant t with probability one or has the Poisson distri- i butio∗n, we write G (n,t) or G (n,Po( )), respectively. Recall that similarly G (n,pˆ) is i i i ∗ ∗ · constructed on the basis of H (n,pˆ) – a hypergraph with independent hyperedges. i In this paper we treat random graphs as random variables. By a coupling (G ,G ) 1 2 of two random variables G and G we mean a choice of a probability space on which a 1 2 random vector (G ,G ) is defined and G and G have the same distributions as G and ′1 ′2 ′1 ′2 1 6 G , respectively. For simplicity of notation we will not differentiate between (G ,G ) and 2 ′1 ′2 (G ,G ). For two graph valued random variables G and G we write 1 2 1 2 G G or G G , 1 2 1 1 o(1) 2 (cid:22) (cid:22) − if there exists a coupling (G ,G ), such that in the probability space of the coupling G 1 2 1 is a subgraph of G with probability 1 or 1 o(1), respectively. Moreover, we write 2 − G = G , 1 2 if G and G have the same probability distribution (equivalently there exists a coupling 1 2 (G ,G ) such that G = G with probability one). 1 2 1 2 Note that, for any λ, in H (n,Po(λ)) each edge appears independently with proba- i bility 1 exp( λ/ n ) (see [8]∗). Thus − − i (cid:0) (cid:1) n (7) G (n,Po(λ)) = G n,1 exp λ . i i ∗ − − i (cid:18) (cid:18) (cid:18) (cid:19)(cid:19)(cid:19) . We gather here a few useful facts concerning couplings of random graphs. For proofs see [13, 14]. Fact 1. Let M be a sequence of random variables and let t be a sequence of numbers. n n (i) If Pr M t = o(1) then G (n,M ) G (n,t ). n n i n 1 o(1) i n (ii) If Pr{ M ≥ t} = o(1) then G∗ (n,t ) (cid:22) − G ∗(n,M ). n n i n 1 o(1) i n { ≤ } ∗ (cid:22) − ∗ Fact 2. Let (G ) and (G ) be sequences of independent random graphs. If i i=1,...,m ′i i=1,...,m G G , for all i = 1,...,m, i (cid:22) ′i then m m G G . i (cid:22) ′i i=1 i=1 [ [ Fact 3. Let G = G (n), G = G (n), and G = G (n) be random graphs. If 1 1 2 2 3 3 G G and G G 1 1 o(1) 2 2 1 o(1) 3 (cid:22) − (cid:22) − then G G . 1 1 o(1) 3 (cid:22) − Fact 4. Let G = G (n) and G = G (n) be two random graphs, such that 1 1 2 2 (8) G G or G G . 1 2 1 1 o(1) 2 (cid:22) (cid:22) − Then for any increasing property A liminfPr G (n) limsupPr G (n) . 1 2 n { ∈ A} ≤ n { ∈ A} →∞ →∞ 7 Proof. Define event := G G on the probability space of the coupling (G ,G ) 1 2 1 2 E { ⊆ } existing by (8). Then for any increasing property A Pr G Pr G Pr 2 2 { ∈ A} ≥ { ∈ A|E} {E} Pr G Pr 1 ≥ { ∈ A|E} {E} = Pr G 1 {{ ∈ A}∩E} = Pr G +Pr Pr G 1 1 { ∈ A} {E}− {{ ∈ A}∪E} Pr G +Pr 1 1 ≥ { ∈ A} {E}− = Pr G +o(1). 1 { ∈ A} The result follows by taking n → ∞ Proof of Theorem 1. We will show only (4) in the case S √S . The remaining cases 3 1 ≫ follow by similar arguments. Here we should note that S = S S and S = Θ(S ). 2 1 3 2 1 − Let w . Denote by V the set of vertices which have chosen feature w (i.e. V = i i i i ∈ W v : w W(v) ). Let i { ∈ V ∈ } V for V 2; i i (9) V = | | ≥ i′ ( otherwise. ∅ For each 1 i m define ≤ ≤ X = V ; i i | | (10) Y = V ; i | i′| Z = I , i Yi isodd { } where I is an indicator random variable of the event A. Note that X , 1 i m, are A i ≤ ≤ independent random variables with binomial distributions Bin(n,p ), 1 i m. Now i ≤ ≤ let Y 3Z i i M = − and M = Z . 2 3 i 2 1 i m 1 i m ≤X≤ ≤X≤ Let [V ]beagraphwiththevertexset andtheedgesetcontaining thoseedgesfrom G i′ V (n,m,p) which have both ends in V (i.e. its edges form a clique with the vertex set V ). G i′ i′ For each 1 i m, we construct independently a coupling of G n, Yi 3Zi G (n,Z ) ≤ ≤ ∗2 −2 ∪ ∗3 i and [V ]. Given Y = y and Z = z , for each i independently, we generate instances of G i′ i i i i (cid:0) (cid:1) G n, yi 3zi and G (n,z ). Let Y = y be the number of non-isolated vertices in the 2 −2 3 i i′ i′ co∗nstructed instance∗of G n, yi 3zi G (n,z ). By definition y y . Set now V to be a(cid:0)union o(cid:1)f the set of n∗o2n–isol−a2ted ∪vert∗i3ces ofiG n, yi 3zi Gi′ ≤(ni,z ) and y i′ y (cid:0) (cid:1) ∗2 −2 ∪ ∗3 i i − i′ vertices chosen uniformly at random from the remaining ones. This coupling implies (cid:0) (cid:1) Y 3Z G n, i − i G (n,Z ) [V ]. 2 3 i i ∗ 2 ∪ ∗ (cid:22) G (cid:18) (cid:19) 8 GraphsG n, Yi 3Zi G (n,Z ), 1 i m, areindependent, and [V ], 1 i m, ∗2 −2 ∪ ∗3 i ≤ ≤ G i ≤ ≤ are independent. Therefore by Fact 2 and the definition of (n,m,p), we have (cid:0) (cid:1) G G (n,M ) G (n,M ) = G n, Yi 3Zi G (n,Z ) ∗2 2 ∪ ∗3 3 ∗2 −2 ∪ ∗3 i 1 i m ≤[≤ (cid:0) (cid:0) (cid:1) (cid:1) (11) [V ] i (cid:22) G 1 i m ≤[≤ = (n,m,p). G By definition m m m m E Y = E (X I ) = S and E Z = E (I I ) = S . i i − Xi=1 1 i Xiisodd − Xi=1 3 i=1 i=1 i=1 i=1 X X X X Moreover m m Var Y = (VarX +VarI 2(EX I EX EI )) i i Xi=1 − i Xi=1 − i Xi=1 i=1 i=1 X X m (EX EI +2EX EI ) ≤ i − Xi=1 i Xi=1 i=1 X m = EY +2(np )2(1 p )n 1 i i i − − i=1 X(cid:0) (cid:1) m EY +3(np np (1 p )n 1) i i i i − ≤ − − i=1 X(cid:0) (cid:1) 4S 1 ≤ and m m Var Z = (VarI +VarI 2(EI I EI EI )) i Xiodd Xi=1 − Xiodd Xi=1 − Xiodd Xi=1 i=1 i=1 X X m (EI EI +2EI EI ) ≤ Xiodd − Xi=1 Xiodd Xi=1 i=1 X m (EX EI +2EX EI ) ≤ i − Xi=1 i Xi=1 i=1 X 4S . 1 ≤ Therefore by Chebyshev’s inequality, for any ω , 1 (ω )2 S /√S , with high proba- ′ ′ 3 1 ≪ ≪ bility m 1 (12) Pr Y S ω S = o(1), i 1 ′ 1 − ≥ ≤ ω ((cid:12) (cid:12) ) ′ (cid:12)Xi=1 (cid:12) p m(cid:12) (cid:12) (cid:12) (cid:12) S 3 Pr (cid:12) Z S (cid:12) ω S = o(1). i 3 ′ 1 ((cid:12) − (cid:12) ≥ ) ≤ S1ω′ (cid:12)Xi=1 (cid:12) p (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 9 Thus with probability 1 o(1) − S 3S 4ω √S 1 3 ′ 1 M − − , 2 ≥ 2 M S ω S . 3 3 ′ 1 ≥ − p Therefore by Fact 1 S 3S 4ω √S G n, 1 − 3 − ′ 1 G n,S ω S 2 3 3 ′ 1 ∗ 2 ∪ ∗ − (cid:18) (cid:19) (cid:16) Gp(n,(cid:17)M ) G (n,M ) (n,m,p). 1 o(1) 2 2 3 3 (cid:22) − ∗ ∪ ∗ (cid:22) G WemayassumethatintheabovecouplingG n, S1 3S3 4ω′√S1 andG n,S ω √S ∗2 − −2 ∗3 3 − ′ 1 are independent. The main reason for this is(cid:16)the fact that eve(cid:17)n though (cid:0)M2 and M3 are(cid:1) dependent (i.e. also G (n,M ) and G (n,M ) are dependent), the choice of a hyper- 2 2 3 3 edge of H (n, ) in a g∗iven draw in th∗e construction of G (n,M ) and G (n,M ) is i 2 2 3 3 ∗ · ∗ ∗ independent from choices in other draws. Moreover note that in the coupling, in order to get G (n,M ) G (n,M ) from a sum of independent graphs G n, S1 3S3 4ω′√S1 ∗2 2 ∪ ∗3 3 ∗2 − −2 ∪ G n,S ω √S we may proceed in the following way. Given M (cid:16)= m and M = m(cid:17) : 3 3 ′ 1 2 2 3 3 ∗ − –(cid:0)if m2 ( or m3(cid:1)resp. ) is larger than (S1 3S3 4ω′√S1)/2 ( or S3 ω′√S1 resp. ) − − − then we make m (S 3S 4ω √S )/2 (or m (S ω √S )) additional draws and 2 1 3 ′ 1 3 3 ′ 1 − − − − − add hyperedges to H (n, S1 3S3 4ω′√S1) ( H (n,S ω √S ) resp. ) ∗2 − −2 ∗3 3 − ′ 1 – if m (m resp.) is smaller than (S 3S 4ω √S )/2 ( or S ω √S resp. ) then 2 3 1 3 ′ 1 3 ′ 1 − − − we delete from H (n, S1 3S3 4ω′√S1) ( H (n,S ω √S ) resp. ) hypredges attributed ∗2 − −2 ∗3 3 − ′ 1 to the last draws to get exactly m , i = 2,3, draws. i Let M and M be random variables with the Poisson distribution 2′ 3′ S 3S 5ω √S 1 3 ′ 1 Po − − and Po S 2ω S , respectively. 3 ′ 1 2 − (cid:18) (cid:19) (cid:16) p (cid:17) Then by sharp concentration of the Poisson distribution S 3S 4ω √S 1 3 ′ 1 Pr M − − = 1 o(1) and Pr M S ω S = 1 o(1). 2′ ≤ 2 − 3′ ≤ 3 − ′ 1 − (cid:26) (cid:27) n p o Therefore by Fact 1, Fact 3 and (7) S 3S 5ω √S S 2ω √S 1 3 ′ 1 3 ′ 1 G n,1 exp − − G n,1 exp − 2 − − 2 n ∪ 3 − − n !! !! 2 3 = G∗2(n,M2′)∪G∗3(n,M3′)(cid:0) (cid:1) (cid:0) (cid:1) S 3S 4ω √S G n, 1 − 3 − ′ 1 G n,S ω S 1 o(1) 2 3 3 ′ 1 (cid:22) − ∗ 2 ∪ ∗ − (cid:18) (cid:19) (cid:16) p (cid:17) (n,m,p). 1 o(1) (cid:22) − G 10