Ramsey numbers of degenerate graphs

Transcription

1 Ramsey numbers of degenerate graphs Choongbum Lee Abstract A graph s d-degenerate f all ts subgraphs have a vertex of degree at most d. We prove that there exsts a constant c such that for all natural numbers d and r, every d-degenerate graph G of chromatc number r has Ramsey number at most 2 d2cr V (G). Ths solves a conjecture of Burr and Erdős from Introducton Ramsey theory studes problems that can be grouped under the common theme that every large system contans a hghly organzed subsystem. A classcal example s the celebrated van der Waerden theorem [34] assertng that n every colorng of the natural numbers wth a fnte number of colors, one can fnd monochromatc arthmetc progressons of arbtrary fnte length. Ths has motvated further results such as Hales-Jewett theorem [21] and Szemeréd s theorem [33] and had a tremendous nfluence on Combnatorcs and related felds. See [20] for a comprehensve overvew of Ramsey theory. For a graph H, the Ramsey number of H, denoted r(h), s defned as the mnmum nteger n such that n every edge two-colorng of K n, the complete graph on n vertces, there exsts a monochromatc copy of H. The name of the feld has ts orgn n a 1930 paper of Frank P. Ramsey [29], who proved that r(k t ) s fnte for all natural numbers t and appled t to a problem of formal logc. In 1935, Erdős and Szekeres [14] brought Ramsey s theorem to a wder audence by dscoverng an nterestng geometrc applcaton. Plenty of varants and applcatons have been found snce then, and now t s consdered as one of the most mportant results n combnatorcs, lyng at the center of nteracton between several felds. There are many fascnatng problems studyng bounds on Ramsey numbers of varous graphs. Erdős and Szekeres, n the paper mentoned above, establshed a recurrence relaton on the Ramsey numbers of complete graphs that mples r(k t ) ( ) 2t 2 t 1 = 2 (2+o(1))t for all natural numbers t. Later, n 1947, Erdős [13], n one of the earlest applcatons of the probablstc method, proved r(k t ) 2 (1/2+o(1))t. These two bounds together show that r(k t ) s exponental n terms of ts number of vertces t. There have been some nterestng mprovements on these bounds [7, 31], but despte a great amount of effort, the constants n the exponents remans unchanged. See the Department of Mathematcs, MIT, Cambrdge, MA Emal: cb lee@math.mt.edu. Research supported by NSF Grant DMS

2 recent survey paper of Conlon, Fox, and Sudakov [10] for further nformaton on graph Ramsey theory. In 1973, Burr and Erdős [4] ntated the study of Ramsey numbers of sparse graphs and conjectured that the behavor of Ramsey numbers of sparse graphs should be dramatcally dfferent from that of complete graphs. A graph G s d-degenerate f all ts subgraphs contan a vertex of degree at most d. Degeneracy s a natural measure of sparseness of graphs as t mples that for all subsets of vertces X, there are fewer than d X edges wth both endponts n X. Burr and Erdős conjectured that for every natural number d, there exsts a constant c = c(d) such that every d-degenerate graph H on n vertces satsfes r(h) cn. Ths s n strkng contrast wth the case of complete graphs where the dependence on number of vertces s exponental. Ths conjecture has receved much attenton and motvated several mportant developments over the past 40 years. For example, the work of Chvátal, Rödl, Szemeréd, and Trotter [5] from 1983 that we wll dscuss below s based on one of the earlest applcatons of the regularty lemma, and fostered further developments such as the blow-up lemma of Komlós, Sarközy, and Szemeréd [22]. Also, Kostochka and Rödl [23] used a prmtve verson of a powerful new tool n probablstc combnatorcs now famously known as the dependent random choce to study a specal case of the conjecture. See the survey paper of Fox and Sudakov [17] for an overvew on hstory and applcatons of ths fascnatng method. Kostochka and Rödl [24] later gave the frst polynomal bound r(h) c d n 2 for all d-degenerate graphs H on n vertces (where c d s a constant dependng on d) usng a dfferent method. The framework of applyng the dependent random choce technque to embed degenerate graphs was poneered by Kostochka and Sudakov [25], who mproved the bound of Kostochka and Rödl to a nearly lnear bound r(h) 2 c d(log n) 2d/(2d+1) n. Later, Fox and Sudakov [15, 16] refned the method to prove that r(h) 2 c d log n n. Furthermore, lnear bounds were establshed for specal cases such as subdvsons of graphs by Alon [1], random graphs by Fox and Sudakov [16], and graphs wth small bandwdth by the author [26]. The conjecture has also been examned for weaker notons of sparseness. Chvátal, Rödl, Szemeréd, and Trotter [5] proved that f H s a graph on n vertces of maxmum degree at most, then r(h) c( )n, where c( ) s a constant dependng only on. Ther proof used the regularty lemma and thus the dependence of c( ) on was of tower type. Ths bound has been mproved snce then, by Eaton [12], Graham, Rödl, and Rucńsk [18, 19], and then by Conlon, Fox, and Sudakov [9] to r(h) c log n. For bpartte graphs, Conlon [8] and ndependently Fox and Sudakov [15] proved that a stronger bound r(h) c n holds. These results are close to best possble snce Graham, Rödl, and Rucńsk [19] constructed bpartte graphs H on n vertces of maxmum degree satsfyng r(h) c n (for a dfferent constant c). Chen and Schelp [6] consdered another measure of sparseness. They defned a graph to be p-arrangeable f there s an orderng v 1,, v n of the vertces such that for any vertex v, ts neghbors to the rght of v have together at most p neghbors to the left of v (ncludng v ). Ths s a measure of sparseness that les strctly between degeneracy and bounded maxmum 2

3 degree. They showed that graphs wth bounded arrangeablty have Ramsey number lnear n the number of vertces, mplyng, n partcular, that the Burr-Erdős conjecture holds for planar graphs. Furthermore, Rödl and Thomas [30] showed that graphs wth no K p -subdvson have arrangeablty less than p 8, and therefore have Ramsey number lnear n the number of vertces. In ths paper, we buld upon these developments and settle the conjecture of Burr and Erdős. We say that a graph G s unversal for a famly F of graphs f t contans all graphs F F as subgraphs. For an edge colorng of a graph, we say that a color s unversal for a famly F f the subgraph nduced by the edges of that color s unversal for F. Theorem 1.1. There exsts a constant c such that the followng holds for every natural number d and r. For every edge two-colorng of the complete graph on at least 2 d2cr n vertces, one of the colors s unversal for the famly of d-degenerate r-chromatc graphs on at most n vertces. Ths settles the conjecture of Burr and Erdős snce all d-degenerate graphs have chromatc number at most d + 1. Moreover, for fxed values of r, Theorem 1.2 s best possble up to the constant n the exponent. To see ths, consder a random graph G on (1 ε)2 d n vertces of densty 1 2 and let H be the complete bpartte graph K d,n d wth d vertces n one part and n d vertces n the other part. It s well-known that n G, wth hgh probablty, every d-tuple of (dstnct) vertces has fewer than (1 ε 2 )n common neghbors. Therefore, G does not contan a copy of H. Snce the complement of G can also be seen as a random graph of densty 1 2, we see that the complement of G does not contan a copy of H as well. Another way to see the tghtness of Theorem 1.2 s by consderng the constructon of Graham, Rödl, and Rucńsk [19] mentoned above. The densty of a graph s defned as the fracton of pars of vertces that form an edge. Most of the prevous results mentoned above n fact provdes a densty-embeddng result for bpartte graphs, sayng that every dense enough graph contans a copy. Note that the Ramsey number result follows from such densty-embeddng result snce n every edge two-colorng of a complete graph, one of the colors must have densty at least 1 2. In ths context, the followng theorem generalzes Theorem 1.1 to a densty-embeddng result. Theorem 1.2. There exsts a constant c such that the followng holds for every natural number d. If G s a graph wth at least α cd n vertces and densty at least α, then t s unversal for the famly of d-degenerate bpartte graphs on n vertces. Note that the complete bpartte graph K d,n d mentoned above has a specfc structure. Namely, every vertex on one sde has bounded degree. Untl now, even ths specal case of the conjecture, bpartte graphs wth one sde of bounded degree, was open. A result of Alon [1] mples the case when the degrees are bounded by two. The correspondng densty-embeddng result was proved by Fox and Sudakov [15], who suggested the general case as an nterestng problem to examne. The theorem below addresses ths specal case of the conjecture and shows that we can mprove Theorem 1.2 and get nearly best possble embeddng results. 3

4 Theorem 1.3. Let d be a natural number and α, ε be postve real numbers satsfyng ε α d(d 2). Let G be a graph on (1 + ε)α d n vertces of densty at least α. Then G s unversal for the famly of bpartte graphs H on n vertces wth a vertex partton W 1 W 2 where all vertces n W 1 have W at most d neghbors n W 2, and 2 d W 2 ( W 2 1) ( W 2 d+1) < 1 + ε. Ths theorem strengthens prevous densty-embeddng results for bpartte graphs wth bounded maxmum degree [8, 15], and s close to beng best possble as can be seen by the example of G beng a random graph and H beng the complete bpartte graph K d,n d dscussed above. Let Q n be the hypercube, whch s the graph wth vertex set {0, 1} n where two vertces are adjacent f and only f they dffer n exactly one coordnate. Theorem 1.3 wth ε = n2 2 and α = 1 n 2 shows that r(q n ) 2 2n +n 2 2 n holds for all suffcently large n. Ths bound slghtly mproves the current best known bound r(q n ) c2 2n (for some constant c) of Conlon, Fox, and Sudakov, proved usng a dfferent approach based on the local lemma [11]. It s conjectured [4] that there exsts a constant c such that r(q n ) c2 n for all n. The proofs of the three theorems above are based on dependent random choce and bulds upon several deas developed through prevous applcatons. Whle the proofs of Theorems 1.1 and 1.2 are techncally nvolved, Theorem 1.3 has a short proof that hghlghts one of the man dfferences n our usage of ths technque as compared to the prevous ones. We thus start by presentng the proof of Theorem 1.3 n Secton 2. In Secton 3, we ntroduce some central concepts and notatons and gve a bref outlne of the proofs of our man theorems. In Secton 4, we develop the embeddng strategy that wll be used. The proofs of Theorems 1.1 and 1.2 wll be gven n Secton 5 except for one key lemma, whch wll be proved n Secton 6. We conclude wth some remarks n Secton 7. Notaton. For an nteger m, defne [m] := {1, 2,..., m} and for a par of ntegers m 1, m 2, defne [m 1, m 2 ] := {m 1,, m 2 }, [m 1, m 2 ) := {m 1,, m 2 1}, (m 1, m 2 ] := {m 1 + 1,, m 2 }, and (m 1, m 2 ) := {m 1 + 1,, m 2 1}. For a set of elements X, we defne X t = X X as the set of all t-tuples n X. Let Q be a d-tuple and Q be a d -tuple of elements n some set. We use Q Q to denote the (d + d )-tuple obtaned by concatenatng Q to the end of Q. Let G = (V, E) be an n-vertex graph. For a vertex x and a set T, we defne deg(x; T ) as the number of neghbors of x n T, and defne deg(x) := deg(x; V ). For a set or an ordered tuple of vertces Q, defne N(Q; T ) := {x T : {x, y} E, y Q} as the set of common neghbors n T of vertces n Q. Defne N(Q) := N(Q; V ). For two sets X and Y, defne E(X, Y ) = {(x, y) X Y : {x, y} E} and e(x, Y ) = E(X, Y ). Furthermore, defne e(x) = 1 e(x,y ) 2e(X, X) as the number of edges n X. Let d(x, Y ) = X Y be the densty of edges between X and Y. For a graph H, an embeddng of H to G s an njectve map f : V (H) V (G) for whch {f(v), f(w)} E(G) whenever {v, w} E(H). A partal embeddng of H to G defned on V V (H) s an embeddng of H[V ] nto G. For V V (H) an extenson to V of a partal embeddng f on V s an embeddng g : H[V V ] G such that g V = f. We often abuse notaton and denote the extended map usng the same notaton f. Throughout the paper, we wll be usng subscrpts such as n x 1.1 to ndcate that x s the 4

5 constant comng from Theorem/Lemma/Proposton One-sde bounded bpartte graphs Fox and Sudakov asked f the Burr-Erdős conjecture holds for bpartte graphs where all vertces n one part have ther degree bounded by d. In ths secton, we answer ther queston by establshng an embeddng theorem for such graphs that strengthens several prevous results of a smlar flavor. Defnton 2.1. Let G be a bpartte graph wth vertex partton V 1 V 2. number α and an ordered d-tuple Q V2 d, we defne the θ-defect of Q as For a postve real ω θ (Q) = { 0 f N(Q) θ θ N(Q) otherwse. We smply wrte ω(q) when θ s clear from the context. For a gven ordered d-tuple of vertces Q, the θ-defect ω θ (Q) defned above captures how the number of common neghbors of Q compares to some prescrbed threshold θ. We gve a penalty to the d-tuples havng only a small number of common neghbors. Informally, f a set has small average θ-defect over d-tuples, then most d-tuples n t wll have at least θ common neghbors. Ths weght functon has been consdered before by Alon, Krvelevch, and Sudakov [2] wth a fxed value of the threshold θ. The followng lemma, based on dependent random choce, shows that we can fnd a set n whch the average defect s small. Lemma 2.2. Let d, t be natural numbers, s be a non-negatve nteger satsfyng t s, and η, ε be postve real numbers satsfyng ε < 1. Let G be a bpartte graph of densty at least α wth vertex partton V 1 V 2. Then there exsts a set A V 2 of sze at least A ε 1/d α t V 2 such that 1 A d Q A d ω θ(q) s ηt 1 ε for all θ ηαd V 1. Proof. Snce ω θ s ncreasng n θ, t suffces to prove the lemma when θ = ηα d V 1. Throughout the proof we wll use ths fxed value of θ, and use the notaton ω θ wthout the subscrpt. Let X V1 t be a t-tuple of vertces chosen unformly at random, and defne A = N(X). Note that E [ A ] = P(v N(X)) = ( ) t deg(v) t V 2 1 deg(v) α t V 2. V 1 V 2 V 1 v V 2 v V 2 v V 2 By convexty, we have E [ A d] α dt V 2 d. Fx a d-tuple Q V d 2 and let N(Q) = γθ. Snce ( P Q A d) = ( ) N(Q) t ( γηα d ) t V 1 = = γ t η t α dt, V 1 V 1 5

6 f γ < 1, then ω(q) s P ( Q A d) = γ t s η t α dt < η t α dt. On the other hand, f γ 1, then ω(q) = 0 by defnton. Hence, E = ( ω(q) s P Q A d) < V 2 d η t α dt. Therefore, Q A d ω(q) s E Q V d 2 A d 1 ε η t ω(q) s εα dt V 2 d. Q A d Choose X so that the random varable on the left-hand sde of the nequalty above becomes at least as large as ts expectaton and let A be A for ths choce of vertces. Then A d εα dt V 2 d, and thus A ε 1/d α t V 2. The other clam follows from A d 1 ε η t Q A d ω(q)s 0. We now prove a densty-embeddng theorem for bpartte graphs wth one part havng bounded maxmum degree. Theorem 1.3 follows from the theorem below by takng ε α d(d 2) snce n every graph on n = n 1 + n 2 vertces of densty α, we can fnd a bpartte subgraph wth n 1 and n 2 vertces n each part, havng densty at least α. Theorem 2.3. Let ε be a postve real number. Let H be a bpartte graph on n vertces wth a vertex partton W 1 W 2 where all vertces n W 1 have at most d neghbors n W 2. Let G be a bpartte graph of densty at least α wth vertex partton V 1 V 2 where V 1 (1 + ε)α d W 1 and V 2 ( 1+ε ε )1/d α 2 W 2. If W 2 d W 2 ( W 2 1) ( W 2 d+1) < 1 + ε, then G contans a copy of H. Proof. Let θ = W 1 and note that θ 1 1+ε αd V 1. Throughout the proof we wll use ω wthout subscrpt wth the understandng that ω = ω θ. Apply Lemma 2.2 wth t 2.2 = 2, s 2.2 = 1, d 2.2 = d, η 2.2 = 1 1+ε, and ε 2.2 = ε 1+ε to fnd a set A V 2 of sze A ( ε 1+ε )1/d α 2 V 2 W 2 for whch Q A d ω(q) ( 1 1+ε )2 1 1 ε/(1+ε) A d = 1 1+ε A d. By addng edges f necessary, we may assume that all vertces n W 1 have exactly d neghbors n W 2. Let φ be an njectve map from W 2 to A chosen unformly at random. For each vertex v W 1, let e v be an ordered d-tuple of vertces obtaned from N(v) by arbtrarly orderng the vertces. Note that E ω(φ(e v )) = E [ω(φ(e v ))] = ω(q)p(q = φ(e v )) v W 1 v W 1 v W 1 Q A d W 1 Q A d ω(q) A ( A 1) ( A d + 1). Snce Q A d ω(q) 1 1+ε A d and A W 2, we have E ω(φ(e v )) 1 W 2 d W ε W 2 ( W 2 1) ( W 2 d + 1) < W 1. v W 1 6

7 Therefore there exsts a partcular choce of φ such that v W 1 ω(φ(e v )) < W 1. Note that φ s trvally a partal embeddng of H to G defned on W 2. We wll now extend φ to W 1. Order the vertces n W 1 as v 1,, v n1 n decreasng order of ω(φ(e v )) (where tes are broken arbtrarly). We wll greedly embed the vertces of W 1 followng ths order. Suppose that we extended φ to {v 1,, v 1 } and for smplcty defne e = e v. Frst assume that ω(φ(e )) 0. Note that ω(φ(e )) v W 1 ω(φ(e v )) < W 1 and therefore ω(φ(e )) < W 1. Hence N(φ(e )) = θ ω(φ(e )) > W 1 W 1 =. Snce we have so far embedded at most 1 vertces of W 1, we can defne φ(v ) as a vertex n N(φ(e )) \ {φ(v 1 ),, φ(v 1 )}. On the other hand f ω(φ(e )) = 0, then N(φ(e )) θ W 1 and therefore we trvally have N(φ(e ))\{φ(v 1 ),, φ(v 1 )} and can defne φ(v ) as a vertex n ths set. Thus we can fnd an embeddng of H to G. 3 Prelmnares 3.1 Decomposng H We start wth the followng smple lemma that decomposes a gven degenerate graph nto manageable peces. Lemma 3.1. Let H be an r-chromatc d-degenerate graph on n vertces. Then there exst a natural number k and dsjont subsets {W (j) } [k],j [r] wth the followng propertes: () k log 2 n, () for all (, j) [k] [r], we have W (j) 2 +1 n, () for all j [r], the set [k] W (j) s an ndependent set, and (v) for all (, j) [k] [r], each vertex n W (j) has at most 4d neghbors n,j [r] W (j ). Proof. Let U 1 = V (H). For each 1, defne U +1 U as the set of vertces havng degree at least 4d n the subgraph H[U ]. Snce H s d-degenerate, there are fewer than d U edges n the subgraph H[U ]. Therefore we have 4d U +1 < 2d U, forcng U +1 < 1 2 U. Snce H has n vertces, we have U < 2 ( 1) n and the process contnues for k log 2 n steps. Defne W = U \ U +1 for all [k]. Consder a proper r-colorng of H usng [r], and for each j [r], defne W (j) as the set of vertces of color j n W. One can easly check that all the condtons hold. 3.2 Defect and average defect We wll work wth the decomposton gven by Lemma 3.1. The man objectve s to fnd a partton {V (j) } of the vertex set of the host graph so that we can embed W (j) to V (j) pece by 7

8 pece. As n the prevous secton, the exstence of such embeddng depends on the average defect of ths partton. In ths subsecton, we ntroduce the central concepts and notatons related to the defect functon. We slghtly generalze Defnton 2.1 so that we consder the number of common neghbors n a specfc set. Defnton 3.2. Let G be a graph. For a postve real number θ, a set T V (G), and an ordered tuple Q of vertces, we defne the θ-defect of Q n T as ω θ (Q; T ) = { 0 f N(Q; T ) θ θ N(Q;T ) otherwse. We may smply wrte ω(q; T ) when θ s clear from the context. One can easly check that monotoncty ω θ (Q; T ) ω θ (Q ; T ) holds for all θ θ, Q Q (as sets), and T T. Note that n the prevous subsecton, we utlzed Lemma 2.2 only for the case s = 1, even though Lemma 2.2 provded a varety of bounds. From now on, t would be crucal to consder other values of s. Defnton 3.3. Let G be a graph and d, s be natural numbers. For a postve real number θ, a set T V (G), and a set Q V (G) d, we defne the s-th moment of the θ-defect of Q n T as µ s,θ (Q; T ) = 1 Q ω θ (Q; T ) s. Q Q We may smply wrte µ s (Q; T ) when θ s clear from the context. The monotoncty of the defect functon ω and the fact that t equals ether 0 or a real number at least 1 mples µ s,θ (Q; T ) µ s,θ (Q; T ) for s s, θ θ, and T T. Note that the concluson of Lemma 2.2 can be re-wrtten as an upper bound on µ s,θ (A d ; V 1 ). Most prevous applcatons of dependent random choce can be consdered as controllng the 0-th moment of the θ-defect functon, whch equals the probablty that a random unform d-tuple n Q has fewer than θ common neghbors, whereas here we wll be consderng hgher moments. We wll focus on the cases when Q = A 1 A d for some (not necessarly dstnct) sets A. The followng proposton provdes a useful relaton between average defects of such product sets. Proposton 3.4. If A 1,, A d, T are vertex subsets, then µ s,θ ( d =1 A ; T ) µ s,θ ( d+1 =1 A ; T ). Proof. Snce ω θ (Q; T ) ω θ (Q ; T ) for all Q Q, we have ω θ (Q; T ) s 1 ω θ (Q {v d }; T ) s A d v d A d Q d 1 =1 A Q d 1 =1 A = 1 A d Q d =1 A ω θ (Q; T ) s. Dvde both sdes of the nequalty by d 1 =1 A to obtan µ s,θ ( d 1 =1 A ) µ s,θ ( d =1 A ). 8

9 The followng proposton shows that the contrbuton of d-tuples wth repeated vertces towards the average defect s of small order magntude. Proposton 3.5. Let A 1,, A d be vertex subsets of some graph all of sze at least m, and let T be a vertex subset. Let Q [d] A be the set of d-tuples (v 1,, v d ) for whch v = v j for some dstnct, j [d]. Then Q Q ω θ(q; T ) s r(r 1) 2m Q d =1 A ω θ(q; T ) s. Proof. For each dstnct a, b [d], defne Q a,b as the set of d-tuples n Q whose a-th and b-th coordnates concde. Note that f Q = (v 1,, v d ) and v d 1 = v d, then ω θ (Q; T ) = ω θ ((v 1,, v d 1 ); T ). Therefore by Proposton 3.4, Q Q d 1,d ω θ (Q; T ) s = Q d 1 =1 A ω θ (Q; T ) s 1 A d Q d =1 A ω θ (Q; T ) s. Snce A m for all [d], the concluson follows by summng up the nequaltes for all pars a, b. Next proposton asserts that for a par of sets V 1 and V 2, f the s-th moment of the average defect of V1 d nto V 2 s small, then all d-tuples n V1 d have many common neghbors n V 2. Proposton 3.6. Let d, s, and θ be natural numbers. For all sets of vertces V 1 and V 2, the number of d-tuples Q V1 d satsfyng N(Q; V 2) < θ s less than µ V 1 d/s s,θ (V1 d; V 2). Proof. If a d-tuple Q V d 1 satsfes N(Q; V 2) < θ V 1 d/s, then by defnton, we have ω θ (Q; V 2 ) s > V 1 d. Therefore the number of such d-tuples s less than µ s,θ (V d 1 ; V 2). We wll use the Proposton above mostly n the case when µ s,θ (V1 d; V 2) < 1. For such cases, all d-tuples Q V1 d satsfy N(Q; V 2) θ. V 1 d/s 3.3 Outlne of proof Let H be a d-degenerate graph and G be a host graph that we would lke to embed H nto. Let {W } [k] be the partton of the vertex set of H obtaned from Lemma 3.1 and suppose that we have dsjont vertex subsets {V } [k] of G. We wll embed W to V one at a tme, startng at = k and n decreasng order of ndex. Suppose that we successfully embedded W +1 to V +1 (and all prevous steps) and are about to embed W to V. Let φ denote the partal embeddng that we have at the moment. For each vertex v W, defne N + (v) = N(v) j> W. For smplcty, we assume that N + (v) = d for all v V (H). We can force the extenson of φ to W to be a homomorphsm by embeddng each vertex v W to a common neghbor of vertces n φ(n + (v)). To obtan an embeddng, we must addtonally guarantee that φ s njectve. Let e v be an (arbtrary) ordered tuple obtaned from N + (v). One way to guarantee that φ s an njectve map s by makng the sum v W ω θ (φ(e v ); V ) (1) 9

10 to be small as n the proof of Theorem 2.3. For smplcty, consder the smple case when all vertces n v W satsfy N + (v) W +1. In ths case, f the embeddng φ W+1 was chosen as a unform map from W +1 to V +1, then we would expect the sum above to be W µ 1,θ (V+1 d ; V ). Hence to make the sum small, we would need the sets {V } [k] to have small average defects n an approprately defned way. However, even f we are gven such subsets {V } [k], the estmate above was based on the assumpton that φ W+1 s a unform random map, whch clearly s not true. Nevertheless, we show that (1) s not too far from µ 1,θ (V+1 d ; V ) by showng that our map s not too far from the random unform map. For example, consder the smple case dscussed above where all vertces n v W j satsfy N + (v) W j+1 for all j [k 1]. For a fxed vertex v W, let w 1,, w d be ts neghbors n N + (v) and let e v = (w 1,, w d ). If the embeddng was chosen unformly at random, then the mage of w j would be chosen n the set V +1 but our algorthm chooses ts mage n N(φ(e wj ); V +1 ) nstead. Hence compared to the random unform map, the expected value of ω θ (φ(e v ); V ) would be larger by a factor V +1 j [d] N(φ(e wj );V +1 ). Note that ether N(φ(e wj ); V +1 ) θ or j [d] V +1 N(φ(e wj ); V +1 ) 1 ( d j [d] V +1 N(φ(e wj );V ) = V +1 θ V +1 N(φ(e wj ); V +1 ) ω θ (φ(e wj ); V ). Therefore ) d 1 V +1 d ( d θ d 1 + ω θ (φ(e wj ); V +1 ) d). j [d] The second term n the summand n expectaton has value µ d,θ (V+2 d ; V +1) and thus f ths quantty s less than 1 and everythng worked as planned, then the rght-hand-sde would be at most ( V+1 θ ) d. Ths means that n expectaton, the sum (1) would be ( V+1 θ ) d µ1,θ (V d +1 ; V ). In 2 general, there would be more dependences between the random varables and t gets trcker to control the events. However the underlyng dea, comparng our algorthm wth the unform random map and measurng ts devaton usng hgher moments of the average defect functon, remans the same. We wll formalze ths dea n Secton 4. 4 Embeddng scheme In ths secton, we develop the embeddng scheme. Let G be a graph wth dsjont vertex subsets {V } [k] and H be a graph wth a vertex partton {W } [k] nto ndependent sets where each vertex n W has at most d neghbors n W +1 W k for all [k 1]. Suppose that natural numbers {θ } [k 1] are gven. For [k 1] and a vertex x W, defne N + (x) = N(x) j [+1,k] W j. Add edges to H f necessary so that N + (x) = d for all vertces x V (H) \ W k. For [k] and x W, let e x be an (arbtrary) ordered d-tuple of vertces formed from N + (x). Defne a random map ψ : V (H) V (G) usng the followng random greedy process 1. Take an njecton from W k to V k unformly at random. 10

11 2. For [k 1], gven a map ψ defned on W +1 W k, we then extend ψ to W. Let x 1, x 2,, x m be the vertces n W n decreasng order of ω θ (ψ(e xj ); V ). 3. After embeddng x 1,, x j 1, embed x j as follows where e j = e xj Defne L j = N(ψ(e j ); V ) \ {ψ(x 1 ),, ψ(x j 1 )} as the set of avalable vertces for x j If N(ψ(e j ); V ) =, then declare falure and halt the process If L j < 1 2 N(ψ(e j); V ), then let ψ(x j ) be a vertex n N(ψ(e j ); V ) chosen unformly at random If L j 1 2 N(ψ(e j); V ), then let ψ(x j ) be a vertex n L j chosen unformly at random. If we only run Steps 3-2 and 3-3 (but never Step 3-1) throughout the embeddng process then the resultng map s a homomorphsm from H to G. As we wll later see, Step 3-1 can easly be ruled out f we have control on the defect functon so one can safely assume that ψ s always an homomorphsm from H to G. Hence ψ wll be an embeddng f t s njectve. Thus t would be crucal to understand when we wll run Step 3-3 nstead of Step 3-2. Even though we consder the defect functon ω θ (Q; T ) for varous dfferent choces of θ, Q, and T, these parameters wll be a functon of the vertex x V (H) that we are about to embed at each step. The motvates the followng defntons that smplfy notatons by removng redundant nformaton. Defnton 4.1. Suppose that we are gven {V } [k], {W } [k], and {θ } [k]. For [k 1] and x W, make the followng defntons. () θ x := θ and V x := V. () ω(x; ψ) := ω θx (ψ(e x ); V x ). () Q x := V 1 V d where W 1 W d s the unque product space contanng e x. (v) µ s (x) := µ s,θx (Q x ; V x ) and µ s := max x V (H) µ s (x). { V (v) γ := max 1, max [k 1] θ }. For x V (H), f µ s (x) s fnte, then there are no d-tuples Q Q x havng N(Q; V x ) =. Hence we wll never run Step 3-1. For a vertex x W, the parameter θ x = θ s the defect that s of nterest when embeddng x to V, as whether we run Step 3-2 or 3-3 depends on N(ψ(e x ); V x ) whch s closely related to ω(x; ψ) = ω θx (ψ(e x ); V x ). Note that the enumeraton of vertces n Step 2 s determned by the values ω(x; ψ). Further note that f the embeddng algorthm chose the mage of x as a random unform vertex n V x, then µ s (x) would be the expected value of ω(x; ψ). We wll later see that γ measures the dfference between the expected value of ω(x; ψ) defned by our random process and by the random unform case. The followng theorem establshes a suffcent condton for ψ to be an embeddng. Theorem 4.2. If V k 2 W k and θ 2 W for all [k 1], then the probablty that ψ does not nduce an embeddng of H to G s at most 2 2d+2 γ 2d µ 4d [k 1] W θ. 11

12 The rest of ths secton focuses on provng Theorem 4.2. The proof of Theorem 1.1 then follows by fndng subsets of vertces of G satsfyng the condton of Theorem Proof of Theorem 4.2 Add a set W k+1 of d vertces to H, make the bpartte graph between W k and W k+1 complete, and between W and W k+1 empty for all [k 1]. Denote the resultng graph as H. Defne N + (x) := N(x) W k+1 for each x W k. Thus now for all x V (H), we have N + (x) = d. Let G be a graph obtaned from G by addng a set V k+1 of d vertces adjacent to all other vertces. Defne θ k = V k and note that ω θk (Q; V k ) = 0 for all Q Vk+1 d. Further note that for all x W k, we have ω θk (x; ψ) = 0 and µ s (x) = 0 regardless of the choce of ψ and s snce e x Wk+1 d. For all vertces x V (H) \ W k, the d-tuple e x never contans a vertex n W k+1. Therefore the sets V k+1 and W k+1 that we added has no effect on the parameters defned above. Throughout ths secton, we wll apply the embeddng scheme defned above to embed H to G. However, we wll slghtly modfy Step 1 as follows: 1. Take a map from W k+1 to V k+1 unformly at random (nstead of a random njecton). We then embed the rest of the graph usng the same algorthm (where we also consder = k n Steps 2 and 3). Note that ψ restrcted to the vertces V (H) has the same dstrbuton as the map prevously defned wthout the sets W k+1 and V k+1. The followng lemma gves a suffcent condton for ψ to be njectve. Lemma 4.3. Let G, H, {V } [k+1], {W } [k+1], and ψ be descrbed as above. Suppose that θ 2 W for all [k]. Let φ : V (H ) V (G ) be a map satsfyng x W ω(x; φ) s 1 2 θ for all [k]. If P(ψ = φ) 0, then φ V (H) s njectve. Proof. As dscussed n Secton 3, we have x W ω(x; φ) x W ω(x; φ) s and therefore t suffces to consder the case s = 1. Let φ : V (H ) V (G ) be a map satsfyng x W ω(x; φ) 1 2 θ for all [k]. Fx [k] and condton on the event that ψ = φ after mappng the vertces W +1,, W k. Let x 1, x 2,, x m be the vertces n W n decreasng order of ω(x j ; φ) and defne e j = e xj for each j [m]. Note that ths s the order that the vertces n W wll be mapped to V. Consder the j-th step. Snce x W ω(x; φ) s fnte, we have N(φ(e j ); V ). Hence we wll not run Step 3-1 when mappng v j. If ω(x j ; φ) = 0, then N(φ(e j ); V ) θ 2 W and we thus we determne ψ(x j ) accordng to Step 3-3. If ω(x j ; φ) 0, then by how we ordered the vertces, we have j ω(x j ; φ) 1 2 θ, and thus θ N(φ(e j ); V ) = ω(x j; φ) θ 2j, from whch t follows that N(φ(e j ); V ) 2j. Snce we embedded at most j 1 vertces of W pror to x j, we determne ψ(x j ) accordng to Step 3-3 when embeddng x j. Snce P(ψ(x j ) = φ(x j )) 0 (condtoned on ψ = φ for all prevous vertces), we see that φ(x j ) s dstnct to all prevously 12

13 mapped vertces. Snce the analyss apples to all steps of the embeddng, we can conclude that φ V (H) s njectve. Lemma 4.3 shows that t s crucal to control the quantty x W ω(x; ψ) s. Note that the expected value of ω(x; ψ) s s µ s (x) f ψ(e x ) were unformly dstrbuted n Q x. Ths was the case n the proof of Theorem 2.3, and thus we were able to conclude that x W ω(x; ψ) s s small qute straghtforwardly. Now our stuaton s more complcated, snce ψ(e x ) s no longer unformly dstrbuted n Q x. We gan control on the sum by comparng our dstrbuton wth the unform dstrbuton. Let ν be the dstrbuton on the set of maps φ : V (H ) V (G ) obtaned by the random greedy embeddng algorthm defned above. For a set of vertces I V (H), we say that ν s a probablty dstrbuton obtaned from ν by neutralzng I f n the random greedy process above, every tme we embed a vertex x I, nstead of followng Steps 3-1, 3-2, or 3-3, we choose the mage of x as a unform random vertex n V x. For example, f we neutralze all vertces, then the resultng dstrbuton s a unform dstrbuton over all maps φ : V (H ) V (G ) satsfyng φ(w ) V for all [k + 1]. From now on, we use ψ to denote a random map from V (H ) to V (G ) whose dstrbuton s determned by the probablty measure under { consderaton. In V contrast, we use φ to denote fxed (non-random) maps. Recall that γ = max 1, max [k] θ }. Lemma 4.4. Let I 1 V (H) be a subset of vertces and let ν 1 be the dstrbuton obtaned from ν by neutralzng I 1. Let X be a random varable dependng only on the mages of vertces n J for some J V (H). Defne I 2 = I 1 J and defne ν 2 as the dstrbuton obtaned from ν by neutralzng I 2. Suppose that t = J \ I 1 1. Then E ν1 [X] 2 t γ t E[X] + 2 2t 1 γ 2t E ν2 [X 2 ] + 1 2t Therefore f X 2 X dentcally holds, then E ν1 [X] 2 2t γ 2t E ν2 [X 2 ] + 1 2t y J\I 1 E ν2 y J\I 1 E ν2 [ ω(y; ψ) 2t ]. [ ω(y; ψ) 2t ]. Note that the frst term on the rght-hand-sde, E ν2 [X 2 ], under the probablty measure ν 2 s determned completely by a set of vertces whose mages are chosen unformly at random. Hence ths lemma allows us to compare the dstrbuton on maps defned by our random greedy algorthm wth the random unform map. 2 V Proof. For each vertex y I 2 \ I 1, defne C y (φ) = y N(φ(e. For a set W V y);v y) (H ) of sze W = t, fx a map φ : W V (G ). We use the notaton ψ t = φ to ndcate the event that the random map ψ obtaned after embeddng the frst t vertces s φ. Defne I W = I 2 \ (I 1 W ). Let ν W be the dstrbuton obtaned from ν 1 by neutralzng I 1 I W. We prove that for all W, t and 13

14 φ as above, E ν1 [X ] ψ t = φ E νw X C y (ψ) ψ t = φ. y I W We prove ths by (reverse) nducton on t. If t = V (H), then I W = and so the random varables on both sdes of the nequalty equals X. Therefore the above trvally holds. Let us now nvestgate the value t whle assumng that the above s true for all larger values. Let W be a set of sze t and φ be a map defned on W. Condtoned on ψ t = φ, we know whch vertex wll be embedded next, say that t s x t+1 W for some [k]. Defne W = W {x t+1 }. For each z V, let φ z be the extenson of φ obtaned by defnng φ(x t+1 ) = z. Then ] [ ] E ν1 [X ψ t = φ = z V P ν1 (ψ t+1 = φ z ψ t = φ) E ν1 X ψ t+1 = φ z. Therefore by the nductve hypothess, we have ] E ν1 [X ψ t = φ P ν1 (ψ t+1 = φ z ψ t = φ) E νw X z V = P ν1 (ψ t+1 = φ z ψ t = φ) E νw X z V y I W y I W C y (ψ) ψ t+1 = φ z C y (ψ) ψ t+1 = φ z, where the second equalty follows snce the dstrbuton of ν W and ν W dffer only on the mage of x t+1 whch s fxed once we condton on ψ t+1 = φ z. If I W = I W, then x t+1 / I 2, and thus P ν1 (ψ t+1 = φ z ψ t = φ) = P νw (ψ t+1 = φ z ψ t = φ). Therefore the rght-hand-sde above s P νw (ψ t+1 = φ z ψ t = φ) E νw X C y (ψ) ψ t+1 = φ z = E νw X C y (ψ) ψ t = φ, z V y I W y I W provng our clam. On the other hand f I W I W, then I W = I W {x t+1 } and v t+1 J \ I 1. In ths case, snce P νw (ψ t+1 = φ z ψ t = φ) = 1 V, we have P ν1 (ψ t+1 = φ z ψ t = φ) Therefore ] E ν1 [X ψ t = φ P ν1 (ψ t+1 = φ z ψ t = φ) E νw X z V 2 N(φ(e xt+1 ); V ) = C x t+1 (φ) P νw (ψ t+1 = φ z ψ t = φ). y I W C xt+1 (φ) P νw (ψ t+1 = φ z ψ t = φ) E νw X z V C y (ψ) ψ t+1 = φ z y I W C y (ψ) ψ t+1 = φ z. 14

15 ( 2 V xt+1 d, Snce C xt+1 (ψ) = N(ψ(e xt+1 );V xt+1 ) and e x t+1 j +1 j) W the value of Cxt+1 (ψ) s determned once we condton on ψ t+1 = φ z. Snce I W = I W {v t+1 }, we have ] E ν1 [X ψ t = φ P νw (ψ t+1 = φ z ψ t = φ) E νw X C y (ψ) ψ t+1 = φ z z V y I W = E νw X C y (ψ). y I W [ Hence we proved the clam. The clam for t = 0 gves E ν1 [X] E ν2 X y I ] 2 \I 1 C y (ψ). Note that C y (ψ) = 2 V y N(ψ(e y);v y) 2 V y max{ 1 θ y, ω(y;ψ) θ y }. Therefore f I 2 \ I 1 = t, then ( ) max{1, ω(y; ψ)} C y (ψ) 2 t V y θ y y I 2 \I 1 y I 2 \I 1 2 t V y 1 (max{1, ω(y; ψ)}) t 2 t γ t 1 (1 + ω(y; ψ) t ). θ y t t y I 2 \I 1 y I 2 \I 1 y I 2 \I 1 Hence for C = 2 t γ t, E ν1 [X] E ν2 [CX] + 1 t E ν2 [CX] + 1 t y I 2 \I 1 E ν2 1 2 y I 2 \I 1 E ν2 [CX] E ν 2 [ C 2 X 2] + 1 2t [ CX ω(y; ψ) t ] ( Eν2 [ C 2 X 2 + ω(y; ψ) 2t]) y I 2 \I 1 E ν2 [ ω(y; ψ) 2t ]. We plan to gan control on the defects ω(x; ψ) by usng Lemma 4.5. As explaned above, the frst term n the rght-hand-sde of Lemma 4.5 gves a drect comparson between our process and the random unform map. However the second term n the rght-hand-sde of Lemma 4.5 s stll problematc snce t s n general determned by vertces that are not yet neutralzed. We repeatedly apply Lemma 4.5 to further gan control on these terms. As we proceed, the set of neutralzed vertces further propagates and eventually there wll be no vertces left to be neutralzed. Lemma 4.5. For all x V (H), we have E [ ω(x; ψ) 2d] 2 2d+1 γ 2d µ 4d. Proof. For a vertex x V, let T 0 be the vertex-weghted rooted tree labelled by vertces n V (H) wth a sngle root vertex labelled x havng weght 1. For j 0, suppose that we constructed a weghted rooted labelled tree T j, and let L j = V (T j ) \ V (T j 1 ). For a node a L j, defne 15

16 y a V (H) as the label of a, and P (a) as the set of vertces on the path from the root to the parent node of a n the tree T j. Let I(a) = W k+1 b P (a) N + (y b ) and let F (a) = N + (y a ) \ I(a). σ(a) For each vertex y F (a), add a chld to a labelled y and let ts weght be 2 F (a), where σ(a) s the weght of a n T j. Let T j+1 be the tree obtaned by dong ths process for all a L j. Note that the process eventually stops snce each path from a root to a leaf has labels of the form (y 1, y 2,, y s ) for vertces y j W j satsfyng 1 < 2 < < s. Let T t be the fnal tree,.e., t s the mnmum nteger for whch T t = T t+1. We may assume that T 1 T 2 T t. Defne γ = 2 2d γ 2d. Let ν be the probablty measure on maps ψ : V (H ) V (G ) nduced by our random embeddng algorthm. For a tree T j and ts node a, let ν a denote the probablty measure obtaned from ν by neutralzng I(a). We clam that for each j 0, E [ω(x; ψ) 2d] σ(a)γ µ 4d + σ(a)e νa [ω(y a ; ψ) 2d ], a L j a V (T j 1 ) where for j = 0 we let T 1 be the empty graph. The root node r s the unque node n T 0 and has weght σ(r) = 1. Moreover, ν r and ν have dentcal dstrbuton snce I(r) = W k+1 and we defned the map from W k+1 to V k+1 to be a random unform map. Therefore the nequalty above holds for j = 0 (n fact equalty holds). Suppose that the clam holds for some j 0. For a vertex a L j, let Γ(a) be the set of chldren of a n T j+1 and let ν 0 be the probablty measure obtaned from ν a by neutralzng the vertces n N + (a). If Γ(a) =, then N + (y a ) I(a) and thus E νa [ω(y a ; ψ) 2d ] µ 2d µ 4d. Otherwse f Γ(a), then for all b Γ(a), we have ν 0 = ν b. Therefore by Lemma 4.5, we have E νa [ω(y a ; ψ) 2d ] γ E ν0 [ω(y a ; ψ) 4d ] Γ(a) b Γ(a) E νb [ω(y b ; ψ) 4d ]. (2) Snce the mages of N + (a) are neutralzed n the measure ν 0, by defnton we have E ν0 [ω(y a ; ψ) 4d ] µ 4d. Snce γ 1, we wll use the same bound (2) for nodes a L j havng Γ(a) = wth the understandng that the second term equals zero for such nodes. Therefore by the nductve hypothess and the fact Γ(a) = F (a) for all a L j, we have E [ω(x; ψ) 2d] σ(a)γ µ 4d + σ(a)e νa [ω(y a ; ψ) 2d ] a L j a V (T j 1 ) a V (T j 1 ) = a V (T j ) σ(a)γ µ 4d + σ(a) γ 1 µ 4d + 2 F (a) a L j σ(a)γ µ 4d + σ(a)e νa [ω(y a ; ψ) 4d ], a L j+1 b Γ(a) E νb [ω(y b ; ψ) 4d ] thus provng the clam. Snce T t = T t+1, n the end we see that E [ ω(x; ψ) 2d] a V (T t) σ(a)γ µ 4d. For each j 0, let U j V (T t ) be the set of nodes of T t that are at dstance j from the root node. Snce 16

17 σ(a) = b Γ(a) 2σ(b) holds for every non-leaf node a V (T t), a smple recursve argument mples a U j 2 j σ(a) = 1, or equvalently a U j σ(a) = 2 j for all j 0. Therefore E [ω(x; ψ) 2d] a V (T t) j [,k] σ(a)γ µ 4d It follows that E [ ω(x; ψ) 2d] 2 2d+1 γ 2d µ 4d. a U j σ(a) 2 2d γ 2d µ 4d = 2 2d γ 2d µ 4d j [,k] Lemma 4.5 can be used to control sums of the form x W ω(x; ψ) s that appear n Lemma 4.3. Usng t, we can prove Theorem 4.2 whch gves a quantfable condton whch wll guarantee the exstence of an embeddng. It s equvalent to the followng form snce θ k = V k. Theorem. If θ 2 W for all [k], then the probablty that ψ does not nduced an embeddng of H to G s at most 2 2d+2 γ 2d µ 4d [k 1] W θ. Proof. Defne λ = x W ω(x; ψ) 2d for each [k]. If [k] λ θ < 1 2, then we have λ < 1 2 θ for all [k], and therefore by Lemma 4.3, ψ nduces an embeddng of H to G. Note that λ k = 0 snce ω(x; ψ) = 0 for all x W k. Thus ψ may not nduce an embeddng of H to G only f [k] λ θ = [k 1] λ θ j.. By Markov s nequalty the probablty of ths event s at [ most 2E [k 1] λ θ ]. By Lemma 4.3, for all x V (H), we have E [ ω(x; ψ) 2d] 2 2d+1 γ 2d µ 4d. Therefore E [λ ] W 2 2d+1 γ 2d µ 4d, and 2E λ θ [k 1] 2 [k 1] 5 Proof of the man theorem 2 2d+1 W θ γ 2d µ 4d 2 2d+2 γ 2d µ 4d [k 1] W θ. In ths secton, we prove Theorems 1.1 and 1.2. We frst prove the bpartte case, Theorem 1.2, for whch a densty-type embeddng result holds and the proof s slghtly more smple. We then prove our man theorem, Theorem 1.1, n Subsecton 5.2. The followng lemma s a slght varant of Lemma 2.2 for non-bpartte graphs. Snce we need to add further condtons to the concluson, nstead of statng the outcome of dependent random choce n terms of a partcular set, we state t n terms of the expected value of the random varables of nterest. We omt ts proof snce t follows from the proof of Lemma 2.2 after makng straghtforward modfcatons. For two vertex subsets X 1 and X 2, we use the notaton e(x 1, X 2 ) to denote the number of pars (v 1, v 2 ) X 1 X 2 that form an edge. Thus e(x, X) = 2e(X) holds for all sets X. 17

18 Lemma 5.1. Let d, s and t be natural numbers satsfyng t s, and η, α be postve real numbers. Let G be a graph wth two sets V 1, V 2 V (G) satsfyng e(v 1, V 2 ) α V 1 V 2. Let X be a t-tuple n V1 t 2). Then for all θ ηα d V 1, we have E[A d ] α dt V 2 d and E V 2 d η t α dt. Q A d ω θ (Q; V 1 ) s The outlne of the proof n both cases, bpartte and non-bpartte, are the same. We frst repeatedly apply Lemma 5.1 to fnd a collecton of r sets {A j } j [r] that have small average defect towards each other. Ths framework was frst developed by Kostochka and Sudakov n [25] and ts varatons have been used n several subsequent work [15, 16, 26]. Although the statement we need straghtforwardly follows from the same proof, we cannot use these lemmas as blackbox snce we need to control hgher moments of the defect functon. The followng proposton summarzes the man observaton that makes ths strategy vable. Proposton 5.2. Let G be a graph wth sets V 1, A 2 V (G). Let X be a t-tuple n A t 2 chosen unformly at random and let A 1 = N(X; V 1 ). Then [ ] E µ s,θ (A d 2; A 1 ) = µ s,θ (A d+t 2 ; V 1 ) Proof. For all Q A d 2, we have N(Q; A 1) = N(Q X; V 1 ). Therefore ω θ (Q; A 1 ) s = ω θ (Q X; V 1 ) s, where we consder Q X as a (d + t)-tuple. Therefore E [ω θ (Q; A 1 ) s ] = E [ω θ (Q X; V 1 ) s ] = ω θ (Q Y ; V 1 ) s P(X = Y ). Snce P(X = Y ) = 1 A 2 t, t follows that [ ] E µ s,θ (A d 2; A 1 ) = E 1 A 2 d Q A d 2 Y A t 2 ω θ (Q; A 1 ) s = The last expresson equals µ s,θ (A d+t 2 ; V 1 ) by defnton. 1 A 2 d+t Q A d Y A 2 t 2 ω θ (Q Y ; V 1 ) s. Once we are gven a collecton of r sets {A j } j [r] that have small average defect towards each other, we use the followng lemma producng sets {V (j) } that s needed for the embeddng scheme of Secton 4. Its proof follows from standard probablstc methods but s rather techncal and wll be gven separately n another secton. For an r-tuples of sets {A j } j [r], we defne A j = j [r]\{j} A j for each j [r]. Lemma 5.3. Let k, d, s, r be fxed natural numbers satsfyng r 2, s 4d and ε, ε be fxed postve real numbers. Let m be a suffcently large natural number dependng on these parameters. Let p for [k] be postve real numbers satsfyng [k] p 1 and p m 1/(10d) for all 18

19 [k]. Suppose that {A j } j [r] are vertex subsets of szes at least εm and at most m satsfyng µ s,θ (A d j ; A j) < 1 (j) 2 for all j [r] for some θ εm. Then there exst sets {V } (,j) [k] [r] satsfyng the followng condtons. Defne θ = 1 2r p θ for all [k]. () For all [k] and j [r], we have V (j) A j and V (j) p A j. () For all ((, j), ( 1, j 1 ),, ( d, j d )) ([k] [r]) d+1 satsfyng j 1,, j d j, we have µ s,θj a [d] V (ja) a ; V (j) max { } ε, 8r d ε d µ s,θ (A d j; A j ). Moreover, f r = 2, then the factor r d ε d can be replaced by 1. () For all (, j) (, j ) the sets V (j) and V (j ) are dsjont. 5.1 Bpartte graphs We frst fnd a par of sets (A 1, A 2 ) for whch the d-tuples n A d 1 have small average defect nto A 2, and vce versa for d-tuples n A 2. Ths wll be acheved by applyng Lemma 5.1 twce. One applcaton wll gve a set A 2 such that the d-tuples n A d 2 have small average defect nto V (G). Now f we apply the lemma agan by choosng the random set X as vertces n A 2, then we wll obtan a set A 1 whose d-tuples have small average defect nto A 2. Proposton 5.2 can be used to show that ths par of sets has the clamed propertes. Lemma 5.4. Let m, d, s, t be fxed natural numbers and α, η be fxed postve real numbers satsfyng t s and η 1 16 α2d. Let G be a bpartte graph of mnmum degree at least αm wth vertex partton V 1 V 2 where V 1 = V 2 = m. Then there exst sets A 1 V 1 and A 2 V 2 satsfyng the followng propertes: () A 1 4 αt m for both = 1, 2, () for all θ 1 2 ηαd+t m, we have µ s,θ (A d 1 ; A 2) 2η t/2 and µ s,θ (A d 2 ; A 1) 2η t/2. Proof. Snce ω θ ω θ holds for all θ θ, t suffces to consder the case when θ = 1 2 ηαd+t m. Throughout the proof, we wll consder ω and µ wth ths fxed value of θ and hence wll omt θ from the subscrpts. Snce θ ηα d+t V 1, we can apply Lemma 5.1 (as n the proof of Lemma 2.2) wth d 5.1 = d + t to fnd a set A 2 V 2 of sze A αt V 2 such that µ s (A d+t 2 ; V 1 ) 2η t. By the mnmum degree condton of G, we know that the subgraph of G nduced on V 1 A 2 has densty at least α. Let X be a t-tuple n A t 2 chosen unformly at random and let A 1 = N(X). Snce θ ηα d A 2, Lemma 5.1 wth (V 1 ) 5.1 = A 2 and (V 2 ) 5.1 = V 1 mples E A 1 d 1 2η t ω(q; A 2 ) s 1 2 αdt V 1 d. Q A d 1 19

20 By Proposton 5.2, t follows that E[µ s (A d 2 ; A 1)] = µ s (A2 d+t ; V 1 ) 2η t. Snce η t/2 1 4 αdt, E A 1 d 1 2η t ω(q; A 2 ) s V 1 d 2η t/2 µ s(a d 2; A 1 ) 1 4 αdt V 1 d. Q A d 1 Let X be a partcular choce of X for whch the random varable on the left-hand-sde becomes at least as large as ts expected value and let A 1 be A 1 for ths choce of X. Frst, A 1 d 1 4 αdt V 1 d mples A αt V 1. Second, A 1 d 1 2η t Q A d 1 ω(q; A 2) s 0 mples µ s (A d 1 ; A 2) 2η t. Thrd, A 1 d V 1 d 2η t/2 µ s (A d 2 ; A 1) 0 mples µ s (A d 2 ; A 1) 2η t/2. The proof of the bpartte case of Burr and Erdős s conjecture follows by combnng Lemma 5.4 wth Lemma 5.3, and then usng the embeddng scheme from Secton 4. Even though the theorem below s stated for graphs of large mnmum degree, the densty-embeddng theorem result follows snce every graph on n vertces of densty at least α contans a subgraph of mnmum degree at least 1 2 αn. Theorem 5.5. For every natural number d and postve real number α, the followng holds for all suffcently large m. If G s a graph on m vertces wth mnmum degree at least αm, then t s unversal for the famly of d-degenerate bpartte graphs on at most d α 48d m vertces. Proof. Defne n = d α 48d m. Let H be a d-degenerate bpartte graph on at most n vertces. By Lemma 3.1, there exsts a vertex partton V (H) = (1) [k] (W W (2) ) satsfyng the followng propertes: () k log 2 n, () for all (, j) [k] [2], we have W (j) 2 +1 n, () both [k] W (1) and [k] W (2) are ndependent sets, and (v) for all (, j) [k] [2], each vertex v W (j) has at most 4d neghbors n W (1) W (2). Defne t = s = 32d and η = 1 32 α12d. Defne θ = 1 8 ηα4d+t m. It s well-known that there exsts a partton V 1 V 2 of V (G) for whch V 1 = V 2 = m 2 and the bpartte subgraph nduced on V 1 V 2 has mnmum degree at least (1 o m (1)) αm 2. Snce η 1 16 (1 o m(1)) 8d α 8d and θ 1 2 η(1 o m(1)) d+t α d+t m 2, Lemma 5.4 wth d 5.4 = 4d and α 5.4 = (1 o m (1))α mples that there exst sets A V 1 and B V 2 satsfyng the followng propertes: () A, B 1 4 (1 + o m(1)) t α t m αt m, and () µ s,θ (A 4d ; B) 2η t/2 and µ s,θ (B 4d ; A) 2η t/2. Defne p = c2 /(20d) where c s a postve constant defned so that [k] p = 1. Then 1 c = 1 [k] 2 /(20d) ( = 1 =0 2 /(20d) 2 1/(20d) ln 2 (ln 2)2 + 20d 2(20d) 2 ) 1 40d.

21 Thus for all [k], we have p p k = c2 k/(20d) c2 log 2 n/(20d) = cn 1/(20d) 1 40d n 1/(20d). Apply Lemma 5.3 to the par of sets (A, B) to obtan parttons A = [k] A and B = [k] B satsfyng the followng condtons for θ = 1 4 p θ: () for all [k], we have A p A and B p ( B, ) () for all (j, 1,, 4d ) [k] 4d+1, we have µ s,θj a [4d] B a ; A j 8µ s,θ (B 4d ; A) 16η t/2, and () for all (j, 1,, 4d ) [k] 4d+1, we have µ s,θj ( a [4d] A a ; B j ) 8µ s,θ (A 4d ; B) 16η t/2. We now apply the embeddng scheme defned n Secton 4. For each [k], we wll map W (1) to A and W (2) to B. For (, j) [k] [2], we wll map the sets W (j) followng the reverse lexcographcal order of (, j). For each set W (j), ts correspondng defect parameter used n the embeddng scheme s θ = 1 4 p max{ A θ, and therefore γ max, B } [k] θ 8m θ 64 By ηα 4d+t. the propertes above, the maxmum average s-th moment defect µ s s at most 16η t/2. For each (, j) [k] [2], we have W (j) = 2 +1 n θ 1 2 c2 /(20d) θ 1280dn 2 /2 ηα 4d+t m d217 n Snce s 16d, we have µ 16d µ s 16η t/2. Therefore [k] j [2] W (j) θ ( ) max{ A, B } 4d µ 16d 2 θ [k] [k] Hence by Theorem 4.2, we can fnd a copy of H n G. 5.2 General graphs 2 /2 α 48d m = /2. 1 ( ) 64 4d 16η t/2 2 2 /2 ηα 4d+t 1 2 /2 224d+4 η t/2 4d α 4d(4d+t) 2 24d+6 (2 5 α 12d ) 12d α 144d2 < 2 8d 2. For general graphs, we prove the lemma correspondng to Lemma 5.4 n two steps. In the frst step we fnd sets A 1, A 2,, A r that have small average defect n one fxed drecton. Lemma 5.6. Let d, s, t, r be natural numbers satsfyng t s and η be a postve real number. In every edge two-colorng of the complete graph K n wth red and blue, n the red graph or the blue graph, there exst dsjont sets of vertces A 1 A r satsfyng the followng condtons: () A j 2 2(t+1)(r 1) n for all j [r], and () for all θ η2 d 2(t+1)(r 1) n, we have µ s,θ (A d j ; A j ) 2ηt for all j < j r. 21

22 Proof. Defne A 2(r 1) = V (G) and arbtrarly color t wth one of the colors. For [2(r 1)], suppose that we constructed a set A of sze at least 2 (t+1)(2r 2 ) n. There exsts a color, say c, of densty at least 1 2 n the set A. Snce θ η2 d A, we may apply Lemma 5.1 (as n the proof of Lemma 2.2) to the subgraph nduced on A wth the edges of color c to fnd a set A 1 A of sze A 1 2 t 1 A such that µ s,θ (A d 1 ; A ) 2η t n the graph consstng of the edges of color c. Color the set A 1 wth the color that we used. Repeat the process to fnd sets A 2r 2 A 2r 3 A 0. Note that A 0 2 (t+1)(2r 2). By the pgeonhole prncple, we can fnd r ndces 1 < < r for whch A j are all colored by the same color, say red. These sets satsfy Property (). Snce µ s,θ (Q; X) µ s,θ (Q; Y ) holds for all sets X Y, we have µ s,θ (A d a ; A b ) 2η t n the red graph for all a < b. Thus the sets A 1, A 2,, A r satsfy the clamed propertes. Gven the sets A 1,, A r constructed n the prevous lemma, we run r more rounds of dependent random choce to produce an r-tuple of sets that have small average defect towards each other. At the -th round, we wll choose a random t -tuple X A t and update each set A j for j to N(X) A j. Ths wll enforce that the average defect of A d nto A s small. Next lemma shows that all the condtons are mantaned throughout ths process. Lemma 5.7. Let d, r, s, and t be fxed natural numbers satsfyng t s. Let ξ = 2 20(d+t) 8r+2 r and θ = ξ 2 n. In every edge two-colorng of the complete graph K n wth red and blue, n the red graph or the blue graph, there exst sets A j for j [r] satsfyng the followng propertes: () A j θ for all j [r], and () µ s,θ (A d j ; A j) ξ t for all j [r]. Proof. For = 0, 1,, r, defne t = 8 r+1 (d + t) and d = d + r j=+1 t j. Note that 1 6 t d t +1 for all [r 1]. Defne ξ = 2 20t0r, θ 0 = ξn, and θ 1 = ξθ 0 = ξ 2 n. We may apply Lemma 5.6 wth d 5.6 = d 0, s 5.6 = 0, t 5.6 = t 0, r 5.6 = r, and η 5.6 = 2 16t 0r snce θ η2 d 0 (t 0 +1)(2r 2). Ths gves sets B 1 B r n, wthout loss of generalty, the red graph, satsfyng B j 2 4t0r n for all j [r], and µ 0,θ0 (B d 0 j ; B j ) 2(2 16t0r ) t 0 ξ t0/2 for all j < j r. Let A 0,j = B j for all j [r]. For each = 0, 1,, r, we wll teratvely construct sets {A,j } j [r] satsfyng the followng propertes: (a) A,j θ 0 for all j r, (b) A,j θ 1 for all j <, (c) µ 0,θ0 (A d,j ; A,j ) ξt /2 for all j < j r, (d) µ s,θ1 (A d, j ; A,j) ξ t /2 for all j. Note that the propertes holds for = 0 (only relevant propertes are (a) and (c)). In the end, the sets A j = A r,j for j [r] satsfy Propertes () and () by Propertes (b) and (d) snce d r = d, θ 1 ξ 2 n and t r = 8(d + t) 2t. 22