aa r X i v : . [ m a t h . P R ] N ov Tight upper tail bounds for cliques
B. DeMarco ∗ and J. Kahn † Abstract
With ξ k = ξ n,pk the number of copies of K k in the usual (Erd˝os-R´enyi) random graph G ( n, p ), p ≥ n − / ( k − and η >
0, we show when k > ξ k > (1 + η ) E ξ k ) < exp h − Ω η,k min { n p k − log(1 /p ) , n k p ( k ) } i . This is tight up to the value of the constant in the exponent.
Let G ( n, p ) be the Erd˝os-R´enyi random graph on n vertices, in which everyedge occurs independently with probability p , and let H be a fixed graphwith v H = | V ( H ) | and e H = | E ( H ) | . A copy of H in G ( n, p ) is any subgraphof G ( n, p ) isomorphic to H . It has been a long studied question (e.g. [5,6, 11, 12, 13, 15, 17]) to estimate, for η > ξ H = ξ n,pH the number ofcopies of H in G ( n, p ), Pr ( ξ H > (1 + η ) E ξ H ) . (1)To avoid irrelevancies, let us declare at the outset that we always assume p ≥ n − /m H , where, as usual (e.g. [10, p.6]), m H = max { e K /v K : K ⊆ H } (2)(so n − /m H is a threshold for “ G ⊇ H ”; see [10, Theorem 3.4]); in particular,when H = K k we assume p ≥ n − / ( k − . For smaller p the problem is not AMS 2000 subject classification: 60F10, 05C80Key words and phrases: upper tails, large deviations, random graphs, subgraph counts* supported by the U.S. Department of Homeland Security under Grant AwardNumber 2007-ST-104-000006. † Supported by NSF grant DMS0701175. η the probability in (1) is easily seen tobe Θ(min { n v K p e K : K ⊆ H, e K > } ); see [10, Theorem 3.9] for a start),and we will not pursue it here.Janson and Ruci´nski [12] offer a nice overview of the methods used priorto 2002 to obtain upper bounds on the probability in (1), by far the morechallenging part of the problem. To get an idea of the difficulty, note thateven for the case that H is a triangle, only quite poor upper bounds wereknown until a breakthrough result of Kim and Vu [15], who used, inter alia ,the “polynomial concentration” machinery of [14] to show, for p > n − log n ,exp p [ O η ( n p )] < Pr( ξ H > (1 + η ) E ξ H ) < exp[ − Ω η ( n p )] . (3)(The easy lower bound, seemingly first observed in [17], is, for example, theprobability of containing a complete graph on something like (1 + η ) / np vertices. Of course the subscript η in the lower bound is unnecessary if,for example, η ≤
1, which is what we usually have in mind.) Polynomialconcentration was also used by Vu [16] to show that if H is strictly balancedand E ξ H ≤ log n , thenPr( ξ H > (1 + η ) E ξ H ) < exp[ − Ω η ( E ξ H )] . (4)The result of [15] was vastly extended in a beautiful paper of Janson,Oleszkiewicz and Ruci´nski [11], where it was shown that for any H and η ,exp p [ O H,η ( M H ( n, p ))] < Pr( ξ H > (1 + η ) E ξ H ) < exp[ − Ω H,η ( M H ( n, p ))] , (5)thus determining the probability (1) up to a factor O (log(1 /p )) in the ex-ponent for constant η . A definition of M is given in Section 10; for now wejust mention that (for p ≥ n − / ( k − ) M K k ( n, p ) = n p k − . (It should alsobe noted that in the limited range where it applies, the upper bound in (4)is better than the one in (5).)While it seems natural to expect that the lower bound in (5) is “usually”the truth (see Section 10 for a precise guess), the only progress in thisdirection until quite recently was [13], which established the upper boundexp[ − Ω( M H ( n, p ) log / (1 /p ))] for H = K or C (the 4-cycle) and some values of p .The log(1 /p ) gap was finally closed for the case H = K by Chatterjee[5] and, independently, the present authors [6]. More precisely, [5] showedthat for a suitable C depending on η and p > Cn − log n ,2r( ξ K > (1 + η ) E ξ K ) < p Ω η ( n p ) , while [6] showed, somewhat more generally, that for p > n − ,exp[ − O η ( f (3 , n, p ))] < Pr( ξ K > (1 + η ) E ξ K ) < exp[ − Ω η ( f (3 , n, p ))] , where f ( k, n, p ) := min { n p k − log(1 /p ) , n k p ( k ) } . (In what follows we willoften abbreviate f ( k, n, p ) = f ( k, n ).)In this paper we considerably extend the method of [6] to settle theproblem for general cliques and a bit more. Theorem 1.1.
Assume H on k vertices has minimum degree at least k − (that is, the complement of H is a matching). Then for all η > and p ≥ n − / ( k − , Pr ( ξ H ≥ (1 + η ) E ( ξ H )) ≤ exp [ − Ω η,H ( f ( k, n, p ))] . Theorem 1.2.
For H = K k and for all p ≥ n − / ( k − , Pr ( ξ H ≥ E ( ξ H )) ≥ exp [ − O H ( f ( k, n, p ))] . Remarks.
1. We are most interested in the “nonpathological” range where f ( k, n, p ) = n p k − log(1 /p ), so when p ≥ n − / ( k − (log n ) / [( k − k − (or abit less). It may be helpful to think mainly of this range as we proceed.2. Though mainly concerned with the case H = K k in Theorem 1.1, weprove the more general statement for inductive reasons. For noncliques thebound of Theorem 1.1 is not usually tight; more precisely: it is tight (upto the constant in the exponent) if p = Ω(1) or if ∆ := ∆ H = k − p = Ω( n − / ∆ ), in which cases our upper bound agrees with the lower boundin (5); it is not tight if ∆ = k − p = o (1) (see the proof of Lemma 2.4)or if H = K k and p < n − c/ ∆ for some fixed c > p = o ( n − / ∆ ) is probably enough here—which would completethis little story—but we don’t quite show this).In the next section we show that Theorem 1.1 follows from an analogousassertion for k -partite graphs; most of the paper (Sections 3-8) is then con-cerned with this modified problem. Section 9 gives the proof of Theorem1.2 and Section 10 contains a few concluding remarks.3 Reduction
For the rest of the paper we set t = log(1 /p ) and take H to be a graph withvertices v , v , . . . , v k . We define G = G ( n, p, H ) to be the random graphwith vertex set V = V ∪ · · · ∪ V k , where the V i ’s are disjoint n -sets andPr( xy ∈ E ( G )) = p whenever x ∈ V i and y ∈ V j for some v i v j ∈ E ( H ), thesechoices made independently. We define a copy of H in G to be a set of vertices { x , . . . , x k } with x i ∈ V i and x i x j ∈ E ( G ) whenever v i v j ∈ E ( H ); use X n,pH for the number of such copies; and set Ψ( H, n, p ) = E ( X n,pH ) = n k p e H . Whenthere is no danger of confusion we will often use X nH —or, for typographicalreasons X ( H, n )—for X n,pH and Ψ( H, n ) for Ψ(
H, n, p ).The next two propositions show an equivalence between G ( n, p ) and G with regard to upper tails for subgraph counts. In each we set α = | Aut( H ) | n k / ( kn ) k ∼ k − k | Aut( H ) | (where as usual ( a ) b = a ( a − · · · ( a − b + 1)). Proposition 2.1.
For η > and ε = η/ (2 + η ) , Pr( X n,pH ≥ (1 + ε )Ψ( H, n, p )) > αε − α + αε Pr( ξ kn,pH ≥ (1 + η ) E ( ξ kn,pH ))We omit the proof of Proposition 2.1 since it is a straightforward general-ization of the case H = K proved in [6]. Proposition 2.2.
For any ε > there is a C = C ε,H such that for p >Cn − /m H , Pr (cid:0) X n,pH ≥ (1 + ε )Ψ( H, n, p ) (cid:1) < ξ kn,pH ≥ (1 + αε/ E ( ξ kn,pH )) . (See (2) for m H .) Proof.
We may choose G ∗ = G ( kn, p ) by first choosing G = G ( n, p, H ) andthen letting E ( G ∗ ) = E ( G ) ∪ S where Pr( xy ∈ S ) = p whenever x = y , x ∈ V i and y ∈ V j for some v i v j E ( H ), these choices made independently. Write ξ and X for the numbersof copies of H in G ∗ and G respectively (thus ξ = ξ kn,pH and X = X n,pH ), andset ξ ∗ = ξ − X . Since E X = α E ξ , we have, using Harris’ Inequality,Pr( ξ > (1 + αε ) E ξ ) ≥ Pr(
X > (1 + ε ) E X ) Pr( ξ ∗ > E ξ ∗ − αε E ξ ); (6)4o we need to say that the second probability on the right is at least 1/2.This is standard, but we summarize the argument for completeness.A result of Janson from [9] (see [10, (2.14)]) givesPr( ξ ∗ ≤ E ξ ∗ − t ) < exp[ − t ] , (7)with ¯∆ = P ∗ σ ∼ τ E I σ I τ ≤ P σ ∼ τ E I σ I τ , (8)where (recycling notation a little) H , . . . are the copies of H in K kn ; I σ = { H σ ⊆ G ∗ } ; “ σ ∼ τ ” means H σ and H τ share an edge (so σ ∼ σ ); and P ∗ means we sum only over σ, τ for which H σ , H τ cannot appear in G .But (very wastefully),¯∆ < n v H X { n v H − v K p e H − e K : K ⊆ H, e K > } < n v H p e H X { n − v K ( Cn − /m H ) − e K : K ⊆ H, e K > } = O ( C − E ξ ) , where C is the constant from (9), which may be taken large compared tothe implied constant in “ O ( · ).” Thus, using (7) with the above bound on ¯∆and t = ( αε/ E ξ , we find that the second probability on the right side of(6) is at least 1 − exp[ − Ω(( αε ) C )] > / viz. Theorem 2.3. If H has minimum degree at least k − , then (a) for all ε > , Pr (cid:0) X n,pH ≥ (1 + ε )Ψ( H, n, p ) (cid:1) < exp [ − Ω H,ε ( f ( k, n, p ))] ;(b) for any τ ≥ , Pr (cid:0) X n,pH ≥ τ Ψ( H, n, p ) (cid:1) < exp[ − Ω H ( f ( k, nτ /k , p ))] . H follows from (a), since (noting that τ Ψ( H, n ) =Ψ(
H, nτ /k ) and using (a) for the second inequality)Pr ( X nH ≥ τ Ψ( H, n )) ≤ Pr (cid:16) X nτ /k H ≥ H, nτ /k ) (cid:17) ≤ exp h − Ω H (cid:16) f ( k, nτ /k , p ) (cid:17)i . We include (b) because it will be needed for induction; that is, for a given H we just prove (a), occasionally appealing to earlier cases of (b).We have formulated the theorem for all p so that the inductive parts ofthe proof don’t require checking that p falls in some suitable range. Note,however, that for the proof we can assume (for our choice of positive con-stants C and c depending on H and ε ) p > Cn − / ( k − , (9)since for smaller p ( > n − /m H ) the theorem is trivial, and p < c, (10)since above this the desired bound is given by (5). As detailed in the nexttwo lemmas, (5), together with some auxiliary results from [11], also allowsus to ignore certain other cases of Theorem 2.3(a). Lemma 2.4. If ∆ H ≤ k − then Pr (cid:0) X n,pH ≥ (1 + ε )Ψ( H, n, p ) (cid:1) ≤ p Ω H,ε ( n p k − ) . Proof.
By Proposition 2.2, it is enough to showPr (cid:0) ξ n,pH ≥ (1 + ε ) E ( ξ n,pH ) (cid:1) ≤ p Ω H,ε ( n p k − ) ; (11)but this follows from (5), which since M H ( n, p ) ≥ n p ∆ H (see [11, Lemma6 . − Ω H,ε ( n p ∆ H )] ≤ exp[ − Ω H,ε ( n p k − t )] . Lemma 2.5.
For any H = K k on k vertices and γ > , if p < n − (1+ γ ) / ( k − then Pr( X nH ≥ (1 + ε )Ψ( H, n )) < p Ω H,ε,γ ( n p k − ) . roof. By Lemma 2.4 we may assume ∆ := ∆ H = k − k − ξ n,pH ≥ (1 + ϑ ) E ( ξ n,pH )) < p Ω ϑ,H ( n p ∆ ) , which, in view of (5) and the definition of M H ( n, p ), will follow if we showthat, for any K ⊆ H , n v K p e K = Ω(( n p ∆ t ) α ∗ K ), or, more conveniently, n v K − α ∗ K p e K − ∆ α ∗ K = Ω( t α ∗ K ) . (12)We need one easy observation from [11] (see their Lemma 6.1): e K ≤ ∆( v K − α ∗ K ) . Then, noting that e K − ∆ α ∗ K < e K < ∆ v K / ≤ ∆ α ∗ K ) and using our upper bound on p , we find thatthe left side of (12) is at least n v K − α ∗ K − (1+ γ )( e K − ∆ α ∗ K ) / ∆ ≥ n v K − α ∗ K − ( v K − α ∗ K )+ γ (∆ α ∗ K − e K ) / ∆ = n γ (∆ α ∗ K − e K ) / ∆ , which (again using (13)) gives (12). This section collects a few standardish large deviation basics that will beused throughout the paper. It’s perhaps worth noting that these elementaryinequalities are the only “machinery” we will need.We use B ( m, α ) for a random variable with the binomial distributionBin( m, α ). The next lemma, which is easily derived from [2, TheoremA.1.12] and [10, Theorem 2.1] respectively (for example), will be used re-peatedly, eventually without explicit mention. Lemma 3.1.
There is a fixed
C > so that for any λ ≤ , K > λ , mand α , Pr( B ( m, α ) ≥ Kmα ) < min { ( e/K ) Kmα , exp[ − Cλ Kmα ] } . (14)7 emark. We may assume
Kmα ≥
1. Thus, if emα c < e/K < α − c and the bound in (14) is at most α (1 − c ) Kmα .The next lemma, an immediate consequence of Lemma 3.1 (and theabove Remark), will also be used repeatedly, usually following a preliminaryapplication of Lemma 3.1 to justify the assumption enq c < Lemma 3.2.
Fix c < and assume enq c < . If S ⊆ V i is random with Pr( x ∈ S ) ≤ q ∀ x ∈ V i , these events independent, then for any T , Pr( | S | ≥ T ) < q (1 − c ) T . We also need the following inequality, which is an easy consequence of,for example, [3, Lemma 8.2].
Lemma 3.3.
Suppose w , . . . , w m ∈ [0 , z ] . Let ξ , . . . , ξ m be independentBernoullis, ξ = P ξ i w i , and E ξ = µ . Then for any η > and λ ≥ ηµ , Pr( ξ > µ + λ ) < exp[ − Ω η ( λ/z )] . In this section we list the steps in the proof of Theorem 2.3(a), filling insome definitions as we go along. The proof proceeds by induction on (say) k + e H , so that in proving the statement for H we may assume its truthfor all graphs with either fewer than k vertices or with k vertices and fewerthan e H edges. The case k = 2 is trivial and k = 3 is the main result of [6],so we assume throughout that k ≥ H in which they appear. The remaining copies are then easilyhandled (in Lemma 4.4) using Lemma 3.3.Here and throughout we use C and C ε for (positive) constants dependingon (respectively) H and ( H, ε ), different occurrences of which will usuallydenote different values. Similarly, we use Ω and Ω ε as shorthand for Ω H andΩ H,ε . We say an event E occurs with large probability (w.l.p.) if Pr( E ) > − exp[ − Ω ε ( n p k − t )], and write “ α < ∗ β ” for “w.l.p. α < β ” (where ε is as in the statement of the theorem). Note that (9) (with a suitable C )guarantees that an intersection of, for example, n w.l.p. events is itself aw.l.p. event, a fact we will sometimes use without mention in what follows.By Lemma 2.4 we may assume ∆ H = k −
1. We reorder the vertices of H so that k − d ( v ) ≥ d ( v ) ≥ . . . ≥ d ( v k ) and if d ( v ) = k − v . We set A = V , B = V , C = V and always take a, b and c to beelements of A, B and C respectively. For disjoint
X, Y ⊆ V we use ∇ ( X, Y )for the set of edges with one end in each of X and Y , and ∇ ( X ) for the setof edges with one end in X . We use N ( x ) for the neighborhood of (set ofvertices adjacent to) a vertex x .For K ⊆ H with vertex set { v i : i ∈ T } ( T ⊆ [ k ]), define a copy of K in G (= G ( n, p, H )) to be a set of vertices { x i : i ∈ T } with x i ∈ V i and x i x j ∈ E ( G ) whenever v i v j ∈ E ( K ). For x , x , . . . , x l vertices belonging todistinct V i ’s we use w K ( x , . . . , x l ) for the number of copies of K containing x , . . . , x l ; when K = H we call this the weight of { x , . . . , x l } . We use H S = H − { v i : i ∈ S } ( S ⊂ [ k ]), and abbreviate H { i } = H i , w H S ( · ) = w S ( · ), w { i } = w i and w ∅ ( · ) (= w H ( · )) = w ( · ).Set ϑ = . ε and define δ by (1 + δ ) k = 2. For x ∈ V and i ∈ [ k ], let d i ( x ) = | N ( x ) ∩ V i | , and set d ( x ) = max { d i ( x ) : i ∈ [ k ] } . Say a vertex x is high degree if d ( x ) > (1 + δ ) np , and a copy of H is type one if contains ahigh degree vertex from A, B or C . Lemma 4.1.
W.l.p. G contains less than ϑ Ψ( H, n ) type one copies of H . Let A ′ , B ′ , C ′ denote the subsets of A, B, C respectively of vertices whichare not high degree. For vertices x, y ∈ G let d j ( x, y ) = | N ( x ) ∩ N ( y ) ∩ V j | and d ( x, y ) = max j ≥ d j ( x, y ). A pair of vertices ( x, y ) is high degree if d ( x, y ) > np / . For k > H is type two if it contains a highdegree pair ( x, y ) belonging to either A ′ × C ′ or B ′ × C ′ ; for k = 4 we don’tneed this, and simply declare that there are no copies of type two. Lemma 4.2.
W.l.p. G contains less than ϑ Ψ( H, n ) type two copies of H . Set s = min { t, n k − p (cid:0) k − (cid:1) } , the two regimes corresponding to the tworanges of f ( k, n, p ) (= n p k − s ). Define w ∗ ( · ) in the same way as w ( · ), butwith the count restricted to copies of H that are not type one or two. Set ζ = (cid:26) k − Ψ( H, n, p ) / ( n p k − s ) if k ≥ H, n, p ) / ( n p s ) if k = 4 (15)and (in either case) say ab ∈ ∇ ( A, B ) is heavy if w ∗ ( a, b ) > ζ. Finally, saya copy of H is type three if it is not type one or two and contains a heavyedge, and type four if it is not type one, two or three. Lemma 4.3.
W.l.p. G contains less than ϑ Ψ( H, n ) type three copies of H . emma 4.4. With probability at least − exp[ − Ω ε ( f ( k, n, p ))] G containsless than (1 + 2 ϑ )Ψ( H, n ) type four copies of H . Of course Theorem 2.3(a) (for k ≥
4) follows from Lemmas 4.1-4.4; theseare proved in the next four sections.
For i ∈ [3] set D ( i ) = { x ∈ V i : d ( x ) > np / } and D ( i ) = { x ∈ V i : np / ≥ d ( x ) > (1 + δ ) np } , and for j ∈ [2] set S j ( i ) = P { d ( x ) : x ∈ D j ( i ) } .We will show Proposition 5.1.
For all ≤ i ≤ ,w.l.p. ∀ x ∈ D j ( i ) , w ( x ) /d ( x ) < (cid:26) n k − p e H − ( k − if j = 12 n k − p e H − k +2( k − / if j = 2and Proposition 5.2.
For all ≤ i ≤ ,w.l.p. S j ( i ) < (cid:26) ϑn p k − if j = 1 kn p k − t if j = 2 . (16)The lemma follows since the number of type one copies of H is at most X x : x high degree w ( x ) < ∗ X i =1 ( S ( i ) · n k − p e H − ( k − + S ( i ) · n k − p e H − k +2( k − / ) < ∗ ϑ Ψ( H, n ) + 2 k Ψ( H, n ) p k − / − t ) < ϑ Ψ( H, n ) , using Propositions 5.1 and 5.2 for the first and second inequalities. Proof of Proposition i and condition on ∇ ( V i ) (thus determining D ( i ) and D ( i )). If d H ( v i ) = k −
1, then for any x ∈ D ( i ), induction givesPr( w ( x ) ≥ H i , d ( x ))) < exp[ − Ω( f ( k − , d ( x )))] , whence (noting Ψ( H i , · ) = Ψ( H , · ))Pr( ∃ x ∈ D ( i ) : w ( x ) ≥ H , d ( x ))) < n exp[ − Ω( f ( k − , np / ))] < p n p k − . (17)10imilarly,Pr( ∃ x ∈ D ( i ) : w ( x ) ≥ H , np / )) < n Pr( X np / H i ≥ H i , np / )) < n exp[ − Ω( f ( k − , np / ))] < p n p k − (18)Note that, here and throughout, we omit the routine verifications of inequal-ities like those in the last lines of (17) and (18).If d ( v i ) = k −
2, then v i v j for some j ∈ [ k ]. We partition V j = P ∪ · · · ∪ P ⌊ /p ⌋ with each P ℓ of size at most (1 + δ ) np , and write w ℓ ( x ) forthe number of copies of H containing x and meeting P ℓ . Noting that hereΨ( H , · ) = p − Ψ( H i , · ) (and w ( x ) = P ℓ w ℓ ( x )), we havePr ( w ( x ) ≥ H , d ( x ))) < Pr( ∃ ℓ w ℓ ( x ) ≥ H i , d ( x ))) < p − exp [ − Ω( f ( k − , d ( x )))]for a given x , so thatPr ( ∃ x ∈ D ( i ) : w ( x ) ≥ H , d ( x ))) < np − exp h − Ω( f ( k − , np / )) i < p n p k − , (19)andPr( ∃ x ∈ D ( i ) : w ( x ) ≥ H , np / )) < np − Pr( X np / H i ≥ H i , np / )) < np − exp[ − Ω( f ( k − , np / ))] < p n p k − . (20)Finally, (17)-(20) imply that w.l.p. w ( x ) /d ( x ) < H , d ( x )) /d ( x ) = 2( d ( x )) k − p e H − ( k − /d ( x ) ≤ n k − p e H − ( k − ∀ x ∈ D ( i )and w ( x ) /d ( x ) < H , np / ) /d ( x ) = 2( np / ) k − p e H − ( k − /d ( x ) ≤ n k − p e H − k +2( k − / ∀ x ∈ D ( i ) . roof of Proposition 5.2. We bound |∇ ( D j ( i )) | , which is, of course, an upperbound on S j ( i ). We first assert that, for any i ∈ [3], w.l.p. | D ( i ) | < ϑnp k − / and | D ( i ) | < np k − t. (21)This will follow from Lemmas 3.1 and 3.2 (so really two applications ofLemma 3.1), a combination we will see repeatedly. For a given i and j theevents { x ∈ D j ( i ) } ( x ∈ V i ) are independent with (using Lemma 3.1)Pr ( x ∈ D ( i )) < k Pr( B ( n, p ) > np / ) < k ( ep / ) np / < p . np / and Pr ( x ∈ D ( i )) < k Pr( B ( n, p ) > (1 + δ ) np ) < exp[ − Ω( np )] . An application of Lemma 3.2 now shows that (21) holds w.l.p.Assume then that (21) holds, and for convenience rename its bounds ϑnp k − / = r and np k − t = u ; we may of course assume r ≥ u ≥ |∇ ( D ( i )) | ≥ ϑn p k − ) < Pr( ∃ T ∈ (cid:0) V ( i ) r (cid:1) : |∇ ( T ) | ≥ ϑn p k − ) < (cid:0) nr (cid:1) Pr( B (( k − rn, p ) ≥ ϑn p k − ) < n r ( e ( k − p / ) ϑn p k − < p Ω ε ( n p k − ) and Pr( |∇ ( D ( i )) | ≥ kn p k − t ) < Pr( ∃ T ∈ (cid:0) V ( i ) u (cid:1) : |∇ ( T ) | ≥ kn p k − t ) < (cid:0) nu (cid:1) Pr( B (( k − un, p ) ≥ kn p k − t ) < n u exp[ − Ω( n p k − t )] < p Ω( n p k − ) , with the third inequality in each case given by Lemma 3.1.12 Proof of Lemma 4.2 (Here we are only interested in k ≥ A ′ , C ′ )-pairs, the argument for ( B ′ , C ′ )-pairs being similar.Let A ′′ be the (random) set of vertices of A ′ involved in high-degree ( A ′ , C ′ )-pairs—that is, A ′′ = { a ∈ A ′ : ∃ c ∈ C ′ d ( a, c ) > np / } —and define C ′′ similarly. We will show that w.l.p. | A ′′ | , | C ′′ | < np k − / (22)and w.l.p. w ( a, c ) < t Ψ( H { , } , (1 + δ ) np ) ∀ ( a, c ) ∈ A ′ × C ′ . (23)Combining these we find that the total weight of high degree ( A ′ , C ′ )-pairsis w.l.p. at most( np k − / ) t Ψ( H { , } , (1 + δ ) np ) < n p k − t Ψ( H { , } , n ) < ϑ Ψ( H, n ) , where the second inequality uses Ψ( H { , } , n ) ≤ n − p − (2 k − Ψ( H, n ) and4 p k − t < ϑ (see (10)). Since, as noted above, the same argument shows thatthe contribution of high-degree ( B ′ , C ′ )-pairs is w.l.p. at most ϑ Ψ( H, n ),the lemma follows.
Proof of (22) . Given ∇ ( C ), the events { a ∈ A ′′ } are independent, withPr (cid:0) a ∈ A ′′ (cid:1) < n ( k −
2) Pr[ B ((1 + δ ) np, p ) > np / ] < n ( k − e (1 + δ ) p / ) np / < p . np / =: q, where we use (9), (10) and k ≥ enq / <
1, Lemma 3.2 gives (22) for A ′′ , and of course the same argument applies to C ′′ . Proof of (23) . Here we have lots of room and just bound max { w ( a ) : a ∈ A ′ } , a trivial upper bound on max { w ( a, c ) : a ∈ A ′ , c ∈ C ′ } . Since d ( a ) < (1 + δ ) np (for a ∈ A ′ ) and v ∼ v ℓ ∀ ℓ ∈ [ k ] \ { , } , Theorem 2.3(b)gives (inductively)Pr[ ∃ a ∈ A ′ w ( a ) ≥ t Ψ( H { , } , (1 + δ ) np )] < n exp[ − Ω( f ( k − , (1 + δ ) npt k − ))] < p Ω( n p k − ) (with verification of the second inequality, which does need (9) at one point,again left to the reader). 13 Proof of Lemma 4.3
This requires special treatment when k = 4; see the beginning of Section 7.2for the reason for the split. In Sections 7.1 and 7.2 we set A ′′ = { a : d i ( a ) ≤ (1 + δ ) np ∀ i ≥ } ⊇ A ′ and define B ′′ similarly. k ≥ For reasons that will be explained as we proceed, we need somewhat differentarguments for large and small values of p . Case np ( k − / ≥ log n . Let C b = { c ∈ C ∩ N ( b ) : d ( b, c ) ≤ np / } and W ( A ) = { a : ∃ b ∈ B ′′ , X c ∈ C b ∩ N ( a ) w ( b, c ) > ζ } ⊇ { a : ∃ b, w ∗ ( a, b ) > ζ } (see (15) for ζ ), and define W ( B ) similarly. Remark.
While it may seem more natural to define W ( A ), W ( B ) in terms of w ( a, b ) or w ∗ ( a, b ), the present definition has the advantage of not dependingon ∇ ( A, B ). We will see something similar in Case 2.The point requiring most work here isw.l.p. | W ( A ) | , | W ( B ) | < ϑnp ( k − / t . (24)Given this, the rest of the argument goes as follows. According to Lemma3.1, (24) implies w.l.p. |∇ ( W ( A ) , W ( B )) | < ϑn p k − (25)(since, given the inequality in (24), |∇ ( W ( A ) , W ( B )) | ∼ B ( m, p ) for some m < ϑ n p k − t ; note the inequalities in (24) and (25) depend on separatesets of random edges). On the other hand, an inductive application ofTheorem 2.3(b) givesw.l.p. w ∗ ( a, b ) < H { , } , (1 + δ ) np ) ∀ a, b (26)(using the fact that we are in Case 1 and noting that d ( a ) > (1 + δ ) np implies w ∗ ( a, b ) = 0).Finally, the combination of (25) and (26) bounds the number of type threecopies of H by ϑn p k − · H { , } , (1 + δ ) np ) < ϑ Ψ( H, n ) . W ( A ). We first showw.l.p. w ( b, c ) < tn k − p e H − (3 k − / =: γ ∀ b ∈ B ′′ and c ∈ C b (27)and w.l.p. w ( b ) < n k − p e H − ( k − ∀ b ∈ B ′′ . (28)These will imply, via Lemma 3.3, that the events { a ∈ W ( A ) } are unlikely,and then (24) will be an application of Lemma 3.2.Each of (27) and (28) is given (inductively) by Theorem 2.3(b), withsmall differences in arithmetic depending on d ( v ) and d ( v ): say we are in(a),(b) or (c) according to whether ( d ( v ) , d ( v )) is ( k − , k − k − , k − k − , k − ∇ ( B ∪ C ) and c ∈ C b , w ( b, c ) isstochastically dominated by X := X ( H { , , } , np / ) in (a) and (c), and bythe sum of ⌊ /p ⌋ copies of X in (b). (For the latter assertion, let ℓ be theindex for which v v ℓ and, recalling that b ∈ B ′′ , partition N ( b ) ∩ V ℓ = V ∪ · · · ∪ V ⌊ /p ⌋ with each block of size at most np / .) Theorem 2.3(b) thusgives the upper bound n ⌊ /p ⌋ exp[ − Ω( f ( k − , np / t / ( k − )] < p Ω( n p k − ) (29)on either Pr( ∃ b ∈ B ′′ , c ∈ C b : w ( b, c ) > t Ψ( H { , , } , np / ))(if we are in (a) or (c)) orPr( ∃ b ∈ B ′′ , c ∈ C b : w ( b, c ) > t ⌊ /p ⌋ Ψ( H { , , } , np / ))(if we are in (b)), the inequality in (29) holding because we are in Case 1.(Note that in (29) the ⌊ /p ⌋ is needed only when we are “in (b),” and theterm involving t only when k = 5.)To complete the proof of (27) it just remains to check that γ (recall thisis the right hand side of (27)) is an upper bound on 2 t Ψ( H { , , } , np / ) ifwe are in (a) or (c), and on 2 t ⌊ /p ⌋ Ψ( H { , , } , np / ) if we are in (b).The proof of (28) is similar. Here, because we are in Case 1, Theorem2.3(b) gives the bound n ⌊ /p ⌋ exp[ − Ω( f ( k − , (1 + δ ) np )] < p Ω( n p k − )
15n Pr( ∃ b ∈ B ′′ w ( b ) > H { , } , (1 + δ ) np )) if we are in (a) or (b), andon Pr( ∃ b ∈ B ′′ w ( b ) > ⌊ /p ⌋ Ψ( H { , } , (1 + δ ) np )) if we are in (c); and it’seasy to check that 2Ψ( H { , } , (1 + δ ) np ) or 2 ⌊ /p ⌋ Ψ( H { , } , (1 + δ ) np ) (asappropriate) is less than 4 n k − p e H − ( k − .Finally we return to (24). Fix (and condition on) any value of E ( G ) \∇ ( A, C ) satisfying the inequalities in (27) and (28). It is enough to showthat, under this conditioning and for any a ,Pr( a ∈ W ( A )) < exp[ − Ω( np ( k − / /t )] =: q, (30)since then Lemma 3.2 implies, using enq / < { a ∈ W ( A ) } are independent, | W ( A ) | < ∗ ϑnp ( k − / t . (The assertion enq / < enq c <
1) imposes the most stringent require-ment on p for Case 1.)For (30) we observe that (28) gives (for any a and b ∈ B ′′ ) E X c ∈ C b ∩ N ( a ) w ( b, c ) = p X c ∈ C b w ( b, c ) ≤ p w ( b ) < n k − p e H − k +2 < ζ/ , whence, using Lemma 3.3 with (27), we havePr( a ∈ W ( A )) < Pr (cid:16) ∃ b ∈ B ′′ X { w ( b, c ) : c ∈ C b ∩ N ( a ) } > ζ (cid:17) < n exp[ − Ω( ζ/γ )] < n exp[ − Ω( np ( k − / /t )] < exp[ − Ω( np ( k − / /t )] . Case np ( k − / < log n . Recall that for very small p —in particular for p in the present range—and H = K k , Theorem 2.3 is contained in Lemma2.5; we may thus assume H = K k . Let H ′ = H − v v and, writing w ′ for w H ′ , set W ( A ) = { a : ∃ b ∈ B ′′ , w ′ ( a, b ) > ζ } ⊇ { a : ∃ b w ∗ ( a, b ) > ζ } , (31)and define W ( B ) similarly. (We could also work directly with w ( a, b ) andavoid the extra definitions; but the present treatment, which we will see16gain below, is more natural in that it allows us to ignore the essentiallyirrelevant ∇ ( A, B ).)The argument here is similar to that for Case 1. We again show thatmembership in W ( A ), W ( B ) is unlikely, leading tow.l.p. | W ( A ) | , | W ( B ) | < log n, (32)which, in view of Lemma 3.1, again givesw.l.p. |∇ ( W ( A ) , W ( B )) | < ϑn p k − . (33)On the other hand we will show, by an argument somewhat differentfrom others seen here,w.l.p. w ∗ ( a, b ) < n k − p ( k − ) ∀ a, b. (34)Combining this with (33) gives Lemma 4.3 (for the present case). Proof of (32). Of course it’s enough to prove the assertion for W ( A ). Wefirst observe thatw.l.p. w ( b ) < t Ψ( H { , } , (1 + δ ) np ) < t log k − n =: m ∀ b ∈ B ′′ ; (35)as elsewhere, this is given by an inductive application of Theorem 2.3(b),which says that, for any b ∈ B ′′ ,Pr( w ( b ) > t Ψ( H { , } , (1 + δ ) np )) < exp[ − Ω( f ( k − , (1 + δ ) npt / ( k − ))] < p Ω( n p k − ) . (Note that for very small p the extra factor t in (35)—which did not appearin (28)—is needed for the final inequality here.)We now condition on E ( G ) \ ∇ ( A ) and assume that, as in (35), w ( b ) 1, Lemma 3.2 gives (32). Remark. Of course (34) is the counterpart of (26) of Case 1 (since H is now K k the two bounds differ only by small constant factors); but for very small p the simple inductive derivation of (26) using Theorem 2.3(b) no longerapplies, since f ( k − , (1 + δ ) np ) may be much smaller than f ( k, n ). Proof of (34). We may assume b ∈ B ′ as otherwise w ∗ ( a, b ) = 0. For i ∈ { , . . . , k } let V ∗ i ( a, b ) = { v ∈ V i : some copy of H on a, b contains v } . We will show thatw.l.p. |∇ ( V ∗ i ( a, b ) , V ∗ j ( a, b )) | < n p k − ∀ i, j, a and b ∈ B ′ . (37)That this gives (34) is essentially a special case of a theorem of N. Alon[1], the precise statement used here (see the proof of Theorem 1.1 in [7])being: an r -partite graph with at most ℓ edges between any two of its partscontains at most ℓ r/ copies of K r .For the proof of (37) we fix a, b and i < j , and think of choosing edgesof G in the order: (i) ∇ ( b, V ∪ · · · ∪ V k ); (ii) ∇ ( V α , V β ) for all 3 ≤ α < β ≤ k except ( α, β ) = ( i, j ); (iii) ∇ ( a, V i ∪ V j ); (iv) ∇ ( V i , V j ). (The remainingedges are irrelevant here.)Let H ′′ = H − v i v j . Since b ∈ B ′ , Lemma 2.5 gives (since we are inCase 2) w H ′′ ( b ) < ∗ H , − v i v j , (1 + δ ) np ) =: m. (38)Let V ∗ i be the set of vertices of V i contained in copies of H ′′ that contain b ,and define V ∗ j similarly.If the bound in (38) holds, then each of V ∗ i , V ∗ j has size at most m n p /s . Here it willbe helpful to work with w rather than w ∗ . We treat (heavy) ab ’s with w ( a, b ) > n p and those with w ( a, b ) ∈ (225 n p /s, n p ] separately.To bound the contribution of edges of the first type, set A ∗ = { a : ∃ b ∈ B ′′ , w ′ ( a, b ) > n p } ⊇ { a : ∃ b ∈ B ′ , w ( a, b ) > n p } (where w ′ is as in the paragraph containing (31)), and define B ∗ similarly.We first show w.l.p. | A ∗ | , | B ∗ | < np / . (39)To see this (for A ∗ , say) we condition on the value of ∇ ( B, C ∪ V ) andconsider Pr( a ∈ A ∗ ). Noting that for any a and b ∈ B ′′ ,Pr( w ′ ( a, b ) ≥ n p ) ≤ Pr( d ( a, b ) > np / )+Pr( w ′ ( a, b ) ≥ n p | d ( a, b ) ≤ np / )(where 5 / / a ∈ A ∗ ) < n [2 Pr( B ((1 + δ ) np, p ) > np / ) + Pr( B ( n p / , p ) > n p )] ≤ p Ω( np / ) + p Ω( n p ) . (40)Since (given ∇ ( B, C ∪ V )) the events { a ∈ A ∗ } are independent, Lemma 3.2now gives (39). (Note that when the second term dominates (40), Lemma3.2 gives A ∗ = ∅ w.l.p.)On the other hand, again using Lemma 3.1, we havePr( ∃ a ∈ A, b ∈ B ′ : w ( a, b ) > n p t ) < n Pr( B ((1 + δ ) n p , p ) > n p t ) < p Ω( n p ) , and combining this with (39) gives X { w ∗ ( a, b ) : w ( a, b ) > n p } < ∗ | A ∗ || B ∗ | n p t < ∗ n p . t ( < ϑn p ) . For ab of the second type (i.e. with w ( a, b ) ∈ (225 n p /s, n p ]), wetake J = 15 np / / √ s , set A J = { a : ∃ b ∈ B ′′ , d ( a, b ) > J } , and define B J similarly. Given ∇ ( B, C ∪ V ) the events { a ∈ A J } are independent with,for each a ,Pr( a ∈ A J ) < n Pr( B ((1 + δ ) np, p ) > J ) < np (1 − o (1)) J/ =: q. e (1 + δ ) np / o (1) < J for the second inequality). Since enq / < J is always at least 15, and is n Ω(1) if p > n − / ),Lemma 3.2 gives | A J | < ∗ √ ϑn p /J. Of course an identical discussion applies to | B J | , so we have | A J || B J | < ∗ ϑsn p and, by Lemma 3.1, |∇ ( A J , B J ) | < ∗ ϑn p . Thus, finally, X { w ∗ ( a, b ) : ab heavy, w ( a, b ) ∈ ( n p /s, n p ] } < ∗ |∇ ( A J , B J ) | n p = ϑn p Case H = K − . Recall that v v is the missing edge and an edge ab isheavy if w ∗ ( a, b ) > H, n, p ) / ( n p s ) = 225 n p /s. We proceed moreor less as in the second part of Case 1.Set J = 15 np/ √ s , A J = { a : ∃ b ∈ B ′′ , d ( a, b ) > J } and B J = { b : ∃ a ∈ A ′′ , d ( a, b ) > J } . Given ∇ ( B, C ∪ V ) the events { a ∈ A J } are independentwith, for each a ,Pr( a ∈ A J ) ≤ n Pr( B ((1 + δ ) np, p ) > J ) < np J/ < p J/ =: q (using Lemma 3.1 and J > ep − / (1+ δ ) np for the second inequality). Since(say) enq / < 1, Lemma 3.2 gives | A J | < ∗ n p /J, and similarly for B J . Since ab heavy at least requires a ∈ A J , b ∈ B J and a ∈ A ′ (and since a ∈ A ′ implies w ( a, b ) < ((1 + δ ) np ) ), this says that thenumber of type three copies of H is at most | A J || B J | ((1 + δ ) np ) < ∗ ( n p /J ) ((1 + δ ) np ) < ϑn p Proof of Lemma 4.4 As earlier, set H ′ = H − v v and w ′ = w H ′ . Let X ′ = P a ∈ A,b ∈ B w ′ ( a, b ).Then X ′ = X H ′ depends only on E ( G ) \ ∇ ( A, B ). Thus X ′ < ∗ (1 + ϑ )Ψ( H ′ , n ) = (1 + ϑ )Ψ( H, n ) /p, (41)where the inequality is given by induction if d ( v ) = k − d ( v ) = k − Y := X a ∈ A,b ∈ B min { w ′ ( a, b ) , ζ } { ab ∈ E ( G ) } ≥ X a ∈ A,b ∈ B w ∗ ( a, b ) { w ∗ ( a,b ) ≤ ζ } . In view of (41) it’s enough to show that under any conditioning on E ( G ) \∇ ( A, B ) for which X ′ < (1 + ϑ )Ψ( H, n ) /p ,Pr( Y > (1 + 2 ϑ )Ψ( H, n )) < exp[ − Ω ϑ ( n p k − s )] (= exp[ − Ω ϑ ( f ( k, n, p ))]) . But under any such conditioning (or any conditioning on E ( G ) \ ∇ ( A, B )),the r.v.’s { ab ∈ E ( G ) } are independent; so, noting E Y ≤ pX ′ < (1+ ϑ )Ψ( H, n )and using Lemma 3.3, we havePr ( Y > (1 + 2 ϑ )Ψ( H, n )) < exp[ − Ω ϑ (Ψ( H, n ) /ζ )] = exp[ − Ω ϑ ( n p k − s )] . Recall here H = K k . Set r = ⌈ E ξ H ⌉ = ⌈ (cid:0) nk (cid:1) p ( k ) ⌉ . Note that we only needto prove Theorem 1.2 for small p , for simplicity say p < n − / ( k − log n ,since above this f ( k, n, p ) = n p k − log(1 /p ) and the theorem is given bythe lower bound in (5). It will thus be enough to show Proposition 9.1. For n − / ( k − ≤ p < n − / ( k − log n , Pr( ξ H = r ) > exp[ − O ( r )] Proof. (This is an easy generalization of the argument for k = 3 given in[6].) The number of sets S of r vertex-disjoint copies of H in K n is21 := ( n ) rk r !( k !) r > (cid:18) n k rk k (cid:19) r . (42)For such an S , let Q S and R S be the events { G contains all members of S } and { S is the set of H ’s of G } . We have Pr( Q S ) = p r ( k ) and will show (forany S ) Pr( R S | Q S ) = exp[ − O ( r )] , (43)whence (using (42))Pr( ξ H = r ) > X S Pr( Q S ) Pr( R S | Q S ) = sp r ( k ) exp[ − O ( r )] > n k p ( k ) rk k ! r exp[ − O ( r )] = exp[ − O ( r )] . For the proof of (43), fix S ; let W be the union of the vertex sets of thecopies of H in S ; and for i = 0 , . . . , k , let T ( i ) be the set of H ’s (in K n )having exactly i vertices outside W . We havePr( R S | Q S ) ≥ (1 − p ) | T (0) | k Y i =1 (cid:16) − p ( i ) +( k − i ) i (cid:17) | T ( i ) | (44)= exp[ − O ( r )] . Here the first inequality is given by Harris’ Inequality [8] (which for ourpurposes says that for a product probability measure µ on { , } E (with E a finite set) and events A i ⊆ { , } E that are either all increasing orall decreasing, µ ( ∩A i ) ≥ Q µ ( A i )), and for the second we can use, say, | T ( i ) | < n i ( rk ) k − i for 0 ≤ i ≤ k . (We omit the easy arithmetic, just notingthat all factors but the last (that is, i = k ) in (44) are actually much largerthan exp[ − O ( r )].) 10 Concluding Remarks Of course the big question is, what is the true behavior of the probability(1) for general H ? We continue to use ξ H for ξ n,pH , and here confine ourselvesto η = 1; that is, we’re interested in Pr( ξ H > E ξ H ). As usual we don’t askfor more than the order of magnitude of the exponent.22ne can show, mainly following the argument of Section 9, that for any K ⊆ H Pr ( ξ H ≥ E ξ H ) > exp[ − O H (Ψ( K, n, p ))] (45)(where, recall, Ψ( K, n, p ) = n v K p e K ). As far as we can see, it could be thatthe truth in (1) is always given by the largest of the lower bounds in (45)and (5). For the latter we (finally) define M H ( n, p ) = (cid:26) n p ∆ H if p ≥ n − / ∆ H min K ⊆ H (Ψ( K, n, p )) /α ∗ K if n − /m H ≤ p ≤ n − / ∆ H (46)(where, as usual, α ∗ is fractional independence number; see e.g. [11] or [4]).This is not quite the same as the quantity M ∗ H ( n, p ) used in [11], but, asshown in their Theorem 1.5, the two agree up to a constant factor; so thedifference is irrelevant here. Conjecture 10.1. For any H and p > n − /m H , Pr ( ξ H ≥ E ξ H ) = exp[ − Θ H (min { min K ⊆ H,e K > Ψ( K, n, p ) , M H ( n, p ) t } )] . (47)(Recall t = log(1 /p ).) We remark without proof (it is not quite obviousas far as we know) that, for a given H , the set of p for which the (outer)minimum in (47) is M H ( n, p ) t is the interval [ p K , K is a smallestsubgraph of H with m K = m H and p K is the unique p for which Ψ( K, n, p ) = M H ( n, p ) log(1 /p ).Conjecture 10.1 gives a different perspective on the observation from [11,Section 8.1] that H = K shows that the lower bound in (5) is not alwaystight. In this case M H ( n, p ) = n p for the full range of p above and, ofcourse, ξ H is just Bin( (cid:0) n (cid:1) , p ); so the upper bound in (5) is the truth. Butin fact (45) shows (with a little thought) that the lower bound in (5) is nottight for any H and sufficiently small p ( > n − /m H ), since for small enough p one of the terms Ψ( K, n, p ) in (47) is o ( M H ( n, p ) t ). What’s special about K is that it is the only (connected) H for which the best lower bound is never given by (5); that is, the minimum in (47) is never M H ( n, p ) t .It also seems interesting to estimatePr( ξ H ≥ γ E ξ H ) (48)when γ = γ ( n ) = ω (1). The present results essentially do this for H = K k and “generic” p ; precisely, Theorem 2.3(b) implies (using a mild variant of23roposition 2.1)Pr( ξ H > τ Ψ( H, n, p )) < exp[ − Ω( f ( k, nτ /k , p ))] , (49)which, for p in the range where f ( k, nτ /k , p ) = n τ /k p k − t , is (up tothe constant in the exponent) the probability of containing a clique of size np ( k − / (2 τ ) /k (provided this is not more than (cid:0) nk (cid:1) ). Of course the trickthat gets Theorem 2.3(b) from Theorem 2.3(a) is general, so results on Con-jecture 10.1 give corresponding upper bounds for (48); but these bounds willnot be tight in general, and at this writing we don’t have a good guess asto the general truth in (48). Acknowledgment. We would like to thank one of the referees for anexceptionally careful reading and for pointing out [16]. References [1] N. Alon, On the number of subgraphs of prescribed type of graphs witha given number of edges. Israel J. Math. (1981), no. 1-2, 116-130.[2] N. Alon and J.H. Spencer, The Probablistic Method , Wiley, New York,2000.[3] J. Beck and W. Chen, Irregularities of Distribution , Cambridge Univ.Pr., Cambridge, 1987.[4] B. Bollob´as, Modern Graph Theory , Springer, New York, 1998.[5] S. Chatterjee, The missing log in large deviations for subgraph counts(2010), http://arxiv.org/abs/1003.3498.[6] B. DeMarco and J. Kahn, Upper Tails for Triangles (2010),http://arxiv.org/abs/1005.4471.[7] E. Friedgut and J. Kahn, On the number of copies of one hypergraphin another Israel J. Math. (1998), 251-256.[8] T.E. Harris, A lower bound on the critical probability in a certainpercolation process, Proc. Cam. Phil. Soc. (1960), 13-20.[9] S. Janson, Poisson approximation for large deviations. Random Struct.Alg. (1990), 221-230. 2410] S. Janson, T. Luczak and A. Ruci´nski, Random Graphs , Wiley, NewYork, 2000.[11] S. Janson, K. Oleszkiewicz and A. Ruci´nski, Upper tails for subgraphcounts in random graphs, Israel J. Math. (2004), 61-92.[12] S. Janson and A. Ruci´nski, The infamous upper tail, Random Structures & Algorithms (2002), 317-342.[13] S. Janson and A. Ruci´nski, The deletion method for upper tail esti-mates, Combinatorica (2004), 615-640.[14] J. H. Kim and V. H. Vu, Concentration of multivariate polynomialsand its applications, Combinatorica (2000), 417-434.[15] J. H. Kim and V. H. Vu, Divide and conquer martingales and the num-ber of triangles in a random graph, Random Structures & Algorithms (2004), 166-174.[16] V.H Vu, On the concentration of multivariate polynomials with smallexpectation, Random Structures & Algorithms , (2000),344-363.[17] V. H. Vu, A large deviation result on the number of small subgraphs ofa random graph, Combin. Probab. Comput.10