Almost-spanning universality in random graphs

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1 Almost-spaig uiversality i radom graphs David Colo Asaf Ferber Rajko Neadov Nemaja Škorić Abstract A graph G is said to be H(, )-uiversal if it cotais every graph o at most vertices with maximum degree at most. It is kow that for ay ε > 0 ad ay atural umber there exists c > 0 such that the radom graph G(, p) is asymptotically almost surely H((1 ε), )- uiversal for p c(log /) 1/. Bypassig this atural boudary, we show that for 3 the same coclusio holds whe p 1 1 log 5. 1 Itroductio Give a family of graphs H, a graph G is said to be H-uiversal if it cotais every member of H as a subgraph (ot ecessarily iduced). Uiversal graphs have bee studied quite extesively, particularly with respect to families of forests, plaar graphs ad graphs of bouded degree (see, for example, [4, 5, 8, 9, 11, 13, 14, 15, 17] ad their refereces). I particular, it is of iterest to fid sparse uiversal graphs. Let H(, ) be the family of all graphs o at most vertices with maximum degree at most. Buildig o earlier work with several authors [2, 5, 6], Alo ad Capalbo [3, 4] showed that there are graphs with at most c 2 2/ edges which are H(, )-uiversal. A simple coutig argumet shows that this result is best possible. The costructio of Alo ad Capalbo is explicit. A earlier approach had bee to study whether radom graphs could be H(, )-uiversal. The biomial radom graph G(, p) is the graph formed by choosig every edge of a graph o vertices idepedetly with probability p. We say that G(, p) satisfies a property P asymptotically almost surely (a.a.s.) if Pr [G(, p) P] teds to 1 as teds to ifiity. The first result o uiversality i radom graphs was proved by Alo, Capalbo, Kohayakawa, Rödl, Ruciński ad Szemerédi [5], who showed that for ay ε > 0 ad ay atural umber there exists a costat c > 0 such that the radom graph G(, p) is a.a.s. H((1 ε), )-uiversal for p c(log /) 1/. I this theorem, some slack is allowed by oly askig that the radom graph cotais subgraphs with (1 ε) vertices. However, oe ca also ask whether the radom graph G(, p) cotais all subgraphs of maximum degree with exactly vertices, that is, whether it is H(, )-uiversal. Because we o loger have ay extra room to maoeuvre, this problem is substatially more difficult to treat tha the almost-spaig versio. Nevertheless, Dellamoica, Kohayakawa, Rödl ad Mathematical Istitute, Oxford OX2 6GG, Uited Kigdom. david.colo@maths.ox.ac.uk. Research supported by a Royal Society Uiversity Research Fellowship. Departmet of Mathematics, Yale Uiversity ad Departmet of Mathematics, MIT. asaf.ferber@yale.edu, ferbera@mit.edu. Istitute of Theoretical Computer Sciece, ETH Zürich, 8092 Zürich, Switzerlad. {readov skoric}@if.ethz.ch. 1

2 Ruciński [14] have show that for ay atural umber 3 there exists a costat c > 0 such that G(, p) is a.a.s. H(, )-uiversal for p c(log /) 1/. The case = 2 was later treated by Kim ad Lee [17], who obtaied similar bouds to those i [14]. These results o embeddig large bouded-degree graphs i the radom graph have proved useful i other cotexts [12, 18]. To give a example, we defie the size Ramsey umber ˆr(H) of a graph H to be the smallest umber of edges m i a graph G which is Ramsey with respect to H, that is, such that ay 2-colourig of the edges of G cotais a moochromatic copy of H. A famous result of Beck [10] states that the size Ramsey umber of the path P with vertices is at most c for some fixed costat c > 0. A extesio of this result to graphs of maximum degree was recetly give by Kohayakawa, Rödl, Schacht ad Szemerédi [18], who showed that there is a costat c > 0 depedig oly o such that if H is a graph with vertices ad maximum degree the ˆr(H) c 2 1/ (log ) 1/. A key compoet i their proof is a embeddig lemma like that used i [5]. Oe could therefore hope to improve this boud by improvig the bouds for uiversality i radom graphs. Here we make some iitial progress o these problems by improvig the theorem of Alo, Capalbo, Kohayakawa, Rödl, Ruciński ad Szemerédi [5] o almost-spaig uiversality as follows. Theorem 1.1. For ay costat ε > 0 ad iteger 3, the radom graph G(, p) is a.a.s. uiversal for the family H((1 ε), ), provided that p 1 1 log 5. This result bypasses a atural barrier, sice (log /) 1/ is roughly the lowest probability at which we ca expect that every collectio of vertices will have may eighbors i commo, a coditio which is extremely useful if oe wishes to embed graphs of maximum degree. O the other had, the lowest probability at which oe might hope that the radom graph G(, p) is a.a.s. H((1 ε), )-uiversal is 2/( +1). Ideed, below this probability, G(, p) will typically ot cotai (1 ε) +1 vertex-disjoit copies of K +1 (see, for example, [16]). Thus, for = 3 our result is optimal up to the logarithmic factor, while for 4 the gap remais. I provig Theorem 1.1, we will make use of a recet result of Ferber, Neadov ad Peter [15] which improves the bouds i [6] ad [14] for bouded-degree graphs which satisfy certai desity ad partitioig properties (see Sectio 2.2). Whe embeddig a graph H H((1 ε), ), we will first fid a subgraph H H suitable for the applicatio of the mai result of [15] by removig all small compoets ad certai short cycles i H. We the embed H, after which we place the deleted pieces i a appropriate way so as to obtai a embeddig of H. 1.1 Notatio For a graph G = (V, E), we deote by v(g) ad e(g) the size of the vertex ad edge sets, respectively. For a vertex v V, we write Γ (i) G (v) := {w V : dist(v, w) = i} for the set of vertices at distace exactly i from v. For simplicity, we let Γ (0) G (v) := {v} ad Γ G(v) := Γ (1) G (v). Furthermore, for a set S V, we defie Γ (i) G (S) := {w V : mi v S dist(v, w) = i}. Similarly, we let B (i) G (v) := (v) be the ball of radius i aroud v i G, i.e., the set of all vertices at distace at most i from i j=0 Γ(j) G v. For a iteger k ad a set of vertices S V, we say that S is k-idepedet if B (k) G (v) (S\{v}) = for every v S, i.e., every two vertices i S are at distace at least k + 1 i G. If there is o risk of ambiguity, we omit G from the subscript. 2

3 Give a (hyper)graph H = (V, E), we say that two edges e 1, e 2 E are idepedet if e 1 e 2 =. For two graphs G ad G, we write G = G if they are isomorphic. A family of subsets A 1,..., A k V, for some iteger k, is a partitio of V if A i A j = for all distict i, j {1,..., k} ad V = k i=1 A i. Note that we allow subsets A i to be empty. Fially, for a iteger k N, we use the stadard otatio [k] := {1,..., k}. We use the stadard asymptotic otatio O, o, Ω ad ω. Furthermore, give two fuctios a ad b, we write a b if a = o(b) ad a b if a = ω(b). 2 Tools ad prelimiaries I this sectio, we preset some tools to be used i the proof of our mai result. 2.1 Probabilistic tools We will use lower tail estimates for radom variables which cout the umber of copies of certai graphs i a radom graph. The followig versio of Jaso s iequality, tailored for graphs, will suffice. This statemet follows immediately from Theorems ad i [7]. Theorem 2.1 (Jaso s iequality). Let p (0, 1) ad cosider a family {H i } i I of subgraphs of the complete graph o the vertex set []. Let G G(, p). For each i I, let X i deote the idicator radom variable for the evet that H i G ad, for each ordered pair (i, j) I I with i j, write H i H j if E(H i ) E(H j ). The, for X = i I X i, µ = E[X] = p e(hi), i I δ = E[X i X j ] = (i,j) I I H i H j (i,j) I I H i H j p e(h i)+e(h j ) e(h i H j ) ad ay 0 < γ < 1, Pr[X < (1 γ)µ] e γ 2 µ 2 2(µ+δ). 2.2 Uiversality for icely partitioable graphs I the followig defiitio, we itroduce a family of graphs that admit a ice partitio. Defiitio 2.2. Let, d ad t be positive itegers ad let ε be a positive umber. The family of graphs F(, t, ε, d) cosists of all graphs H o vertices for which there exists a partitio W 0,..., W t of V (H) such that (i) W t = ε, (ii) W 0 = Γ(W t ), (iii) W t is 3-idepedet, 3

4 (iv) W i is 2-idepedet for every 1 i t 1, ad (v) for every 1 i t ad for every w W i, w has at most d eighbors i W 0... W i 1. The followig result, due to Ferber, Neadov ad Peter [15], shows that for a appropriate p a typical G G(, p) is F(, t, ε, d)-uiversal. Theorem 2.3 (Theorem 4.1 i [15]). Let ad t = t() be positive itegers, let d = d() 2 be a iteger ad let ε = ε() < 1/(2d). The the radom graph G(, p) is a.a.s. F(, t, ε, d)-uiversal, provided that p ε 1 t 1/d log 2. We remark that this result is actually stroger tha we eed, sice it gives a statemet about spaig graphs while i this paper we oly deal with almost-spaig graphs. Thus, whe applyig Theorem 2.3, we will make our graph spaig by addig a certai umber of isolated vertices. 2.3 Uiversality for graphs of small size I this sectio, we prove auxiliary lemmas which will allow us to igore small compoets i the proof of Theorem 1.1 (see Phase III i the proof of Theorem 1.1). Lemma 2.4. Let 3 ad k be itegers ad let H H(log k, ) be a coected graph with v(h) +2. The G G(, p) cotais H with probability 1 e ω(), provided that p 2/( +1). Proof. Let (h 1,..., h v(h) ) be a arbitrary orderig of the vertices of H ad let V 1,..., V v(h) V (G) be disjoit subsets of order / log k. We wish to use Jaso s iequality to prove the lemma. For that, we will restrict our attetio to the caoical copies of H: the family {H i } i I cosists of all those copies of H i K with the property that the vertex h j belogs to the set V j for every j {1,..., v(h)}. We ow estimate the parameters µ ad δ defied i Theorem 2.1. Let X be the umber of copies H i, i I, that appear i G. The µ = E[X] satisfies the boud ( ) v(h) ( ) v(h) µ = log k p e(h) log k p v(h) /2 where the last iequality follows from v(h) + 2. Next, ote that for ay proper subgraph J H we have e(j) ( ) v(j) 2 if v(j), ad e(j) 1 2 ((v(j) 1) + 1) = 1 2 (v(j) 1) otherwise. ( ) v(h) log k /( +1), The secod estimate follows from the fact that H is coected ad therefore there exists at least oe vertex i J with degree at most 1. We ca ow rewrite δ as δ = p e(h i)+e(h j ) e(h i H j ) = p 2e(H) e(j). J H H i H j =J (i,j) I I H i H j Usig the observatios above, we split the sum based o the size of v(j), gettig δ v(j) p 2e(H) ( 2 ) + p 2e(H) (v(j) 1)/2. J H H i H j =J J H H i H j =J v(j) v(j) +1 4

5 We ca boud the umber of summads by first decidig o a embeddig of J, which ca be doe i (/ log k ) v(j) ways, ad the o a embeddig of the remaiig parts of the two copies of H which itersect o J (at most (/ log k ) 2(v(H) v(j)) ways), yieldig δ ( ) ( ) v(h) 2v(H) j v(h) 1 ( ) ( ) v(h) 2v(H) j j log k p 2e(H) (j 2) + j log k p 2e(H) (j 1)/2. j=2 j= +1 Fially, by pullig µ 2 outside, we obtai ( ) ( ) δ µ 2 v(h) j v(h) 1 ( ) ( ) v(h) j j log k p (j 2) + j log k p (j 1)/2. j=2 j= +1 By substitutig for p, we get the followig upper boud o the first sum, δ 1 := ( ) ( ) v(h) j j log k p (j 2) (log ) 2kj j j(j 1) j=2 j=2 = (log ) 2 k Proceedig similarly for the secod sum, we have δ 2 := v(h) 1 j= +1 ( ) ( v(h) j log k ) j p (j 1)/2 Sice j + 1 i the above sum, we have j=2 v(h) 1 j= (log ) 2 k j=2 +1 (j 1) j +j +1 (log ) 2 k 1 1/( +1) 1/. (log ) 2jk j j 1 j+1 +1 = 1 j +1 (log ) 2k(j+1) 1, thus it easily follows that δ 2 = o(1/). Summig up, we get δ = o(µ 2 /). Fially, by applyig Theorem 2.1 with parameters µ ad δ, we obtai Pr[X < µ/2] e µ2 /(8(µ+δ)). Sice µ, this implies the coclusio of the lemma. v(h) 1 +1 = j= +1 (log ) 2jk j+1 The ext lemma deals with graphs o at most + 1 vertices. Sice we treat as a costat, it is a stadard applicatio of Jaso s iequality ad we omit the proof. Lemma 2.5. Let 3 be a iteger ad H ay graph o at most +1 vertices. The G G(, p) cotais H with probability 1 e ω(), provided that p 2/( +1). Fially, we make use of Lemmas 2.4 ad 2.5 to show that every large iduced subgraph of G(, p) cotais all coected graphs from H(log k, ) simultaeously. Lemma 2.6. Let ε > 0 be a costat ad 3 ad k be itegers. The, for p 2/( +1), G G(, p) a.a.s. has the followig property: for every V V (G) of order V ε, G[V ] cotais every coected graph H H(log k, )

6 Proof. Let G G(, p), with p as stated i the lemma. By Lemmas 2.4 ad 2.5, for a fixed subset V V (G) of order V ε ad a coected graph H H(log k+1 (ε), ), we have Pr[H G[V ]] = 1 e ω(). Note that as log k log k+1 (ε), these estimates also apply for every coected graph H H(log k, ). Sice there are at most 2 choices for V ad at most log k v H /2 v H =2 e H =1 ( ) v 2 H e H ( log log 2k 2k ) log k = o(2 ) coected graphs H H(log k, ), a applicatio of the uio boud completes the proof. 2.4 Systems of disjoit represetatives i hypergraphs The followig lemma will allow us to place a set of short cycles i the proof of Theorem 1.1 (see Phase II i the proof of Theorem 1.1). We make o effort to optimize the logarithmic factor i the boud o the edge probability p. Lemma 2.7. Let ε > 0 be a costat, 3, 3 g 2 log ad t ε/(32 log 3 ) be itegers ad let D [] be a subset of size ε/(4 log ). The G G(, p) satisfies the followig with probability at least 1 o(1/), provided that p ( log 7 / ) 1/( 1) : for ay family of subsets {Wi,j } (i,j) [t] [g], where (i) W i,j V (G) \ D ad W i,j = 2 for all (i, j) [t] [g], ad (ii) W i,j W i,j = for all i i, there exists a family of cycles {C i = (c i1,..., c ig )} i [t], each of legth g, such that (i) V (C i ) G[D] ad V (C i ) V (C i ) =, for all i i, ad (ii) W i,j Γ G (c ij ) for all (i, j) [t] [g]. Lemma 2.7 will follow as a corollary of Lemma 2.9 ad the followig geeralizatio of Hall s matchig criterio due to Aharoi ad Haxell [1]. Theorem 2.8 (Corollary 1.2, [1]). Let g be a positive iteger ad H = {,..., H t } a family of g-uiform hypergraphs o the same vertex set. If, for every I [t], the hypergraph i I H i cotais a matchig of size greater tha g( I 1), the there exists a fuctio f : [t] t i=1 E(H i) such that f(i) E(H i ) ad f(i) f(j) = for i j. The followig lemma allows us to fid the matchigs required by Theorem 2.8 i a greedy way, i.e., edge by edge. Lemma 2.9. Let ε > 0 be a costat, 3, 3 g 2 log ad k ε/(32 log 3 ) be itegers ad let D [] be a subset of order ε/(4 log ). The G G(, p) satisfies the followig with probability at least 1 o(1/ 2 ), provided that p ( log 7 / ) 1/( 1) : for ay family of subsets {Wi,j } (i,j) [k] [g], where 6

7 (i) W i,j V (G) \ D ad W i,j = 2 for all (i, j) [k] [g], ad (ii) W i,j W i,j = for all i i, ad ay subset D D of order D D g 2 k, there exists i {1,..., k} ad a cycle C = (c 1,..., c g ) G[D ] of legth g such that W i,j Γ G (c j ) for all j {1,..., g}. Proof. Our aim is to show that for a subset D D ad a family {W i,j } (i,j) [k] [g] satisfyig properties (i) ad (ii), the graph G G(, p) fails to satisfy the coclusio of the lemma with probability at most e ω(k log3 ). Sice we ca choose the family {W i,j } (i,j) [k] [g] i at most ( kg 2) 2 ( 2)kg log = 2 o(k log3 ) ways ad D i at most ( g k) 2 g 2 k log 2 4k 2 log3 ways, the lemma follows by a simple applicatio of the uio boud. It remais to prove the desired boud o the probability of a failure. We first itroduce some otatio. Give a cycle C = (c 1,..., c g ) K of legth g, we defie the graph C i by V (C i) = V (C) g W i,j, ad E(C i) = E(C) j=1 j [g] v W i,j {c j, v}. Furthermore, let V 1,..., V g D be arbitrarily chose disjoit subsets of order ε/(16 log 2 ) (this is possible sice D ε/(8 log )), defie the family of caoical cycles C as ad set C := {C = (c 1,..., c g ) C is a cycle ad c j V j for all j [g]} C + := {C i C C ad i [k]}. Observe that if G cotais ay graph from C +, the G cotais the desired cycle. Usig Jaso s iequality, we upper boud the probability that this does ot happe. I the remaider of the proof, we will estimate the parameters µ ad δ defied i Theorem 2.1. Note that each graph C + C + appears i G with probability p g+( 2)g = p ( 1)g. Therefore, ( ) ε g µ = C + p ( 1)g = k 16 log 2 p ( 1)g k log 3. Next, we wish to show that δ = o(µ 2 /k log 3 ). By defiitio, we have δ = p e(c i)+e(c j) e((c i) (C j)). i,j [k] C,C C C i C j We cosider the cases i j ad i = j separately. First, if C i C j for i j ad C, C C, the we have (C i) (C j) = C C. Let J := C C ad observe that e(j) 1, as otherwise C i ad C j would ot have ay edges i commo. Let J 1 be the family cosistig of all possible graphs of the form C C, J 1 := {J = C C C, C C ad e(j) 1}. 7

8 We ca ow estimate the cotributio of such pairs to δ as follows: δ 1 = p e(c i)+e(c j) e(c C ) = i j i j k 2 C,C C e(c C ) 1 J J 1 ( ε 16 log 2 ) 2(g v(j)) p 2( 1)g e(j) = µ 2 J J 1 C,C C C C =J J J 1 ( ε 16 log 2 p 2( 1)g e(j) ) 2v(J) p e(j). Sice e(j) = 1 for v(j) = 2 ad e(j) v(j) otherwise, we ca boud the last sum by J J 1 ( ) ε 2v(J) 16 log 2 p e(j) ( ) ε 4 16 log 2 p 1 + ( ) ε 2v(J) 16 log 2 p v(j). J J 1 J J 1 v(j)=2 v(j)>2 Observe that there are at most ( g v J ) ( ε/(16 log 2 ) ) v J J J 1 ( ) ε 2v(J) 16 log 2 p e(j) (2 log ) 2 ( 16 log 2 ε v J >2 v J graphs J J 1 o v J vertices. Thus, we have ( ) ( ) g ε log 2 p 1 + ( ) ( ) g ε vj v v J >2 J 16 log 2 p v J ) 2 1/( 1) + ( ) ( 2 log ε 16 log 2 ) 1 vj 1/( 1) 32 ε 1 k log 3, where we used that 3. Therefore, we obtai δ 1 = o(µ 2 /k log 3 ). Let us ow cosider the case C i C i, for some i [k] ad distict cycles C, C C. Let J := C C ad observe that v(j) 1 ad v(j) g 1. As before, let J 2 be the family cosistig of all possible graphs of the form C C, J 2 := {J = C C C, C C ad v(j) {1,..., g 1}}. Note that if V (J) V q = {v} for some q {1,..., g}, the {v, w} E(C i C i) for all w W i,q. Therefore, we have e(c i C i) = e(j) + v(j)( 2). With these observatios i had, we ca boud the cotributio of such pairs to δ as follows: δ 2 = i [k] k J J 2 C,C C v(c C ) 1 ( ε 16 log 2 p e(c i)+e(c i) e(c i C i) = i [k] ) 2(g v(j)) p 2( 1)g (e(j)+v(j)( 2)) µ2 k J J 2 C,C C C C =J J J 2 p 2( 1)g (e(j)+v(j)( 2)) ( ε 16 log 2 ) 2v(J) p (e(j)+v(j)( 2)). Sice J is a subgraph of a cycle, we have e(j) v(j). Therefore, we ca boud the last sum by J J 2 ( ) ε 2v(J) 16 log 2 p (e(j)+v(j)( 2)) ( ) ε 2v(J) 16 log 2 p v(j)( 1). J J 2 8

9 Note that there are at most ( g v J ) ( ε/(16 log 2 ) ) v J J J 2 ( ) ε 2v(J) 16 log 2 p v(j)( 1) g 1 v J =1 g 1 v J =1 graphs J J 2 o v J vertices. Thus, we have ( g v J ) ( ) ε vj 16 log 2 p v(j)( 1) ( 2 log 16 log2 ε ) vj log 7 1 log 3. Therefore, we obtai δ 2 = o(µ 2 /(k log 3 )). Fially, we have δ δ 1 +δ 2 = o(µ 2 /(k log 3 )) ad Theorem 2.1 gives the desired upper boud o the probability that G does ot cotai ay graph from C +, completig the proof of the lemma. Proof of Lemma 2.7. Let G G(, p) with p as stated i the lemma. For each i [t], we defie a g-uiform hypergraph H i := (D, E i ) as follows: a set of vertices {v 1,..., v g } D forms a hyperedge if ad oly if G[{v 1,..., v g }] cotais a cycle C = (c 1,..., c g ) such that W i,j Γ G (c j ) for all j [g]. Observe that the existece of a fuctio f with properties as i Theorem 2.8, applied to the family H := {,..., H t }, implies the existece of the desired family of cycles. Therefore, it is sufficiet to prove that, with probability at least 1 o(1/), for every I [t] the hypergraph i I E(H i) cotais a matchig of size at least g I. Sice the requiremets o the family {W i,j } (i,j) [t] [g] are the same as i Lemma 2.9, it follows from the uio boud that, with probability at least 1 o(1/), G satisfies the property give by Lemma 2.9 for D, g ad every k [t]. Let I [t] be a subset of size k for some k [t] ad set D := D. We ow apply the followig procedure kg times: usig the property give by Lemma 2.9 for D, there exists a hyperedge e i I E(H i) such that e D, ad set D := D \ e. Sice e = g ad we repeat the procedure kg times, we have D D kg 2 throughout the whole process. Thus, we ca ideed apply Lemma 2.9 i each step. Furthermore, sice every edge is vertex disjoit from the previously obtaied edges, we have costructed a matchig i i I E(H i) of size kg. As I [t] was chose arbitrarily, this cocludes the proof of the lemma. 3 Proof of Theorem 1.1 Our proof strategy goes as follows. Give a graph H H((1 ε), ), we first remove vertices belogig to small coected compoets from H, writig for the resultig graph. Workig i, we the remove carefully chose iduced cycles of legth at most 2 log (agai, we remove vertices) i such a way that the resultig graph H 2 belogs to the family of graphs F((1 ε ), Θ(log 3 ), ε, 1), for some parameters ε ad ε tedig to zero with ε. Now, usig Theorem 2.3, we fid a embeddig of H 2. The, usig Lemma 2.7, we place the removed cycles ito G i a appropriate way. Fially, usig Lemma 2.6, we complete the embeddig of H by embeddig small compoets oe by oe. We will ow give a formal descriptio of this procedure. Preparig the graph G. Fix some ε > 0 ad iteger 3. Let R, D 3,..., D 2 log [] be arbitrarily chose disjoit subsets of {1,..., } such that R = (1 ε/2) ad D i = ε/(4 log ) for each i {3,..., 2 log }. Let G be a graph with the followig properties: (i) the iduced subgraph G[R] is F((1 ε/2), ( 2 + 1)q + 1, ε, 1)-uiversal, where q = 65ε 1 log 3 ad ε = mi{1/(2 ), ε/(2 ε)}, 9

10 (ii) for every subset V V (G) of order V ε, the iduced subgraph G[V ] cotais every coected graph from the family H(log 4, ), ad (iii) G satisfies the property give by Lemma 2.7 for every g {3,..., 2 log }, t ε/(32 log 3 ) ad D = D g. Observe that by Theorem 2.3 ad Lemmas 2.6 ad 2.7, G G(, p) satisfies properties (i) (iii) asymptotically almost surely, provided that p 1/( 1) log 5. We remark that the boud o p here is determied by Theorem 2.3. Preparig the graph H. Let H H((1 ε), ) ad let H be the subgraph which cosists of all coected compoets of H with at least log 4 vertices. The followig observatio plays a crucial role i our argumet. Claim 3.1. For every vertex v, at least oe of the followig properties hold: (a) B (log ) (v) cotais a vertex w with deg H1 (w) 1, or (log ) (b) [B (v)] cotais a cycle of legth at most 2 log. Proof. Let us assume the opposite, i.e., for every vertex w B (log ) (log ) (v) we have deg H1 (w) = ad (log ) [B (v)] cotais o cycle of legth at most 2 log. The [B 3, it cotais at least log j=1 ( 1)j > vertices, which is clearly a cotradictio. (v)] is a tree ad, sice Let I V ( ) be a maximal (64ε 1 log 3 )-idepedet set i. Write I a for the set of all vertices i I which satisfy property (a) of Claim 3.1 ad set I b := I \I a. Furthermore, for each v I b, (log ) let C v be a cycle of smallest legth i [B (v)], let l v deote its legth ad fix a arbitrary orderig (c 1 v, [..., c lv v ) of the vertices alog C v. By miimality, C v is a iduced cycle. Fially, let ] H 2 := \ v I V (C b v) ad ote that the (64ε 1 log 3 )-idepedece of I implies (3 log ) B (v) V (H 2 ) = { (3 log ) B (v), for v I a, (3 log ) B (v) \ V (C v ), for v I b. (1) Phase I: Embeddig H 2 ito G[R]. We claim that there exists a embeddig of H 2 ito G[R]. Let H 2 be a graph o (1 ε/2) vertices obtaied from H 2 by addig isolated vertices. Usig property (i) of the graph G, i order to show that there exists a embeddig of H 2 ito G[R], it will suffice to prove that H 2 F((1 ε/2), ( 2 + 1)q + 1, ε, 1), where q = 65ε 1 log 3 ad ε = mi{1/2, ε/(2 ε)}. We prove this by fidig a partitio W 0, W 1,..., W ( 2 +1)q+1 of V (H 2 ) with the followig properties: (i) W ( 2 +1)q+1 = ε (1 ε/2), (ii) W 0 = Γ H 2 (W ( 2 +1)q+1), (iii) W ( 2 +1)q+1 is 3-idepedet (i H 2 ), (iv) W i is 2-idepedet (i H 2 ) for every 1 i ( 2 + 1)q, ad 10

11 (v) for every 1 i ( 2 + 1)q + 1 ad for every w W i, w has at most 1 eighbors i W 0... W i 1. First, ote that H 2 cotais at least ε/2 isolated vertices as V (H 2) (1 ε). Sice ε ε/(2 ε) or equivaletly ε (1 ε/2) ε/2, we ca set W ( 2 +1)q+1 to be a set of ε (1 ε/2) isolated vertices. The W 0 = ad W ( 2 +1)q+1 is trivially 3-idepedet. Furthermore, let S q V (H 2 ) \ W ( 2 +1)q+1 be the set of all remaiig vertices i H 2 with degree at most 1 i H 2 ad observe that for each v I we have (3 log ) S q B (v). (2) For v I a, this follows from (1) ad the defiitio of the set I a. For v I b, we have from (1) (3 log ) (3 log ) ad B (v) 3 log > V (C v ) that B (v) V (H 2 ). Thus there exists a vertex (3 log ) w B (v) V (H 2 ) adjacet to some vertex i C v, ad clearly deg H2 (w) 1. Next, for each i {1,..., q 1}, we defie S q i := Γ (i) H 2 (S q ). We first show that S 1,..., S q, W ( 2 +1)q+1 is a partitio of V (H 2 ). Sice disjoitess follows from the costructio, it suffices to prove that for each w V (H 2 ) \ W ( 2 +1)q+1 we have B (q 1) H 2 (w) S q. This ca be see as follows. Sice I is a maximal (64ε 1 log 3 )-idepedet set i, for each vertex w V (H 2 ) \ I we have B (64ε 1 log 3 ) (w) I. Otherwise, we could exted I, cotradictig its maximality. Thus, from (2), we coclude that for each w V (H 2 )\I we have B (q 1) (w) S q. Let us ow cosider the shortest path i from w to a vertex s S q, ad deote the vertices alog such a path by w = p 0, p 1, p 2,..., p q = s, for some q q 1. If p 1,..., p q V (H 2 ), the clearly s B (q 1) H 2 (w). Otherwise, let i be the smallest idex such that p i / V (H 2 ). But the deg H2 (p i 1) 1 ad thus, by defiitio, p i 1 S q, which agai implies B (q 1) H 2 (w) S q. This shows that S 1,..., S q, W ( 2 +1)q+1 is ideed a partitio of V (H 2 ). Furthermore, by costructio, for each i {1,..., q 1} ad each vertex v S i, v has at least oe eighbor i q j=i+1 S j ad thus at most 1 eighbors i i 1 j=1 S j. However, the sets S i are ot ecessarily 2-idepedet i H 2. This ca be fixed i the followig way. The square of H 2, deoted by (H 2 )2, has maximum degree at most 2. Therefore, (H 2 )2 ca be partitioed ito 2 +1 sets L 1,..., L 2 +1 which are idepedet i (H 2 )2 ad thus 2-idepedet i H 2. Now, by settig W (i 1)( 2 +1)+j := S i L j for every i {1,..., q} ad j {1,..., 2 + 1}, we obtai a partitio of V (H 2 ) satisfyig properties (i) (v). To coclude, we have show that H 2 F((1 ε/2), ( 2 + 1)q + 1, ε, 1). Thus, by property (i) of the graph G, there exists a embeddig f : V (H 2 ) R of H 2 ito G[R]. Phase II: Embeddig removed cycles. Cosider some g {3,..., 2 log } ad let I g I b be the set of all vertices v I b such that l v = g. For each (v, j) I g [g], let W v,j := f(γ H (c j v) V (H 2 )). Note that, by costructio, the family of subsets {W v,j } (v,j) Ig [g] satisfies requiremets (i) ad (ii) of Lemma 2.7 with D = D g. Thus, i order to apply Lemma 2.7, it suffices to show that I g is ot too large. The followig claim provides a upper boud o I, ad thus o I g, which i this case suffices. Claim 3.2. I ε 32 log 3. 11

12 Proof. Let u, v I be two distict vertices from I. The B (32ε 1 log 3 ) (u) B (32ε 1 log 3 ) (v) =, as otherwise we would have dist(u, v) 64ε 1 log 3 ad the set I would ot be (64ε 1 log 3 )- idepedet. O the other had, for every v I we have that either Γ (i) (v) for every i {1,..., 32ε 1 log 3 } or B (32ε 1 log 3 ) (v) cotais the whole coected compoet of v, which is of order at least log 4. I either case, we have B (32ε 1 log 3 ) (v) 32ε 1 log 3 ad the desired boud o I follows. Therefore, by property (iii) of the graph G, there exists a family {(c v,1,..., c v,g )} v Ig of vertex disjoit cycles i G[D g ] such that settig f(c j v) := c v,j for every (v, j) I g [g] defies a embeddig of ] H 2 [ v I C g v ito G[R D g ]. Sice this holds for every 3 g 2 log ad the sets D 3,..., D 2 log are disjoit, we obtai a embeddig of ito G. It is worth remarkig that the boud o I, which facilitates the applicatio of Lemma 2.7, is the reaso why we treat small coected compoets separately. Phase III: Embeddig small compoets. As a last step, we have to exted our embeddig of to a embeddig of the whole graph H. Usig the facts that H is of order (1 ε) ad each compoet of H which is ot i is of order at most log 4, we ca greedily embed these compoets oe by oe as follows. Cosider oe such compoet ad let V V (G) be the set of vertices which are ot a image of some already embedded vertex of H. The V ε ad, by property (ii) of the graph G, G[V ] cotais a embeddig of the required compoet. Repeatig the same argumet for each compoet which has ot yet bee embedded, we obtai a embeddig of the graph H. 4 Cocludig remarks The observat reader will have oticed that our argumet does ot apply whe = 2. I this case, oe caot hope to show that uiversality holds all the way dow to p 1/( 1) = 1. Eve to fid a collectio of (1 ε) 3 disjoit triagles, the probability must be at least 2/3. Sice every graph with maximum degree 2 is a disjoit uio of paths ad cycles, it is ot too hard to use argumets similar to those i Sectio 2.3 to show that for ay ε > 0 there exists a costat c > 0 such that if p c 2/3, the radom graph G(, p) is a.a.s. H((1 ε), 2)-uiversal. Our proof relies heavily o the fact that the graphs we are hopig to embed are almost spaig rather tha spaig. I particular, we either kow how to complete the removed cycles or how to add small compoets back ito the graph without makig heavy use of the almost-spaig coditio. Give this, it seems likely that a spaig aalogue of Theorem 1.1 will require ew ideas. We have already metioed that a modificatio of the embeddig techiques from Alo et al. [5] was used by Kohayakawa, Rödl, Schacht ad Szemerédi [18] to prove a subquadratic boud o the size Ramsey umber of bouded-degree graphs. It would be iterestig to kow whether our embeddig techique might be useful for improvig this boud. Ackowledgemets. We would like to thak the aoymous referees for their thorough reviews ad valuable remarks. 12

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