Embedding degenerate graphs of small bandwidth

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1 Embeddng degenerate graphs of small bandwdth Choongbum Lee Abstract We develop a tool for embeddng almost spannng degenerate graphs of small bandwdth. As an applcaton, we extend the blow-up lemma to degenerate graphs of small bandwdth, the bandwdth theorem to degenerate graphs, and make progress on a conjecture of Burr and Erdős on Ramsey number of degenerate graphs. 1 Introducton An embeddng of a graph H nto a graph G s an njectve map f : V H) V G) for whch {fv), fw)} s an edge of G whenever {v, w} s an edge of H. The study of suffcent condtons whch force the exstence of an embeddng of H nto G s a fundamental topc n graph theory that has been studed from varous dfferent aspects. For example, Turán s theorem s a foundatonal result that establshes the mnmum number of edges needed to guarantee an embeddng of a complete graph nto another graph. Another fundamental result s Drac s theorem 1 whch asserts that every n-vertex graph G of mnmum degree at least n contans a Hamlton cycle,.e., a cycle passng through every vertex of the graph exactly once. Drac s theorem nfluenced the development of the study of embeddng large subgraphs graphs whose number of vertces have the same order of magntude as that of the host graph), a classcal example beng Hajnal and Szemeréd s theorem assertng that every graph of mnmum degree at least 1 1 r )n contans n r vertex-dsjont copes of K r. In 1997, Komlós, Sárközy, and Szemeréd 3 made a major breakthrough n ths drecton. They developed the powerful blow-up lemma and used t to tackle varous problems ncludng Posá-Seymour conjecture on power of Hamlton cycles 4, and Alon-Yuster conjecture on F - packng 5. Ths lne of research culmnated n the so called Bandwdth Theorem of Böttcher, Schacht, and Taraz 3. The bandwdth of a graph H s the mnmum nteger b for whch there exsts a labellng of ts vertces by ntegers 1,,..., V H) where j b holds for every edge {, j}. Confrmng a conjecture of Bollobás and Komlós, they proved that for every nteger r and postve real δ, there exsts β such that every n-vertex graph G wth large n) of mnmum degree at least 1 1 r + δ)n contans all n-vertex r-chromatc graphs of bandwdth at most βn as a subgraph. Department of Mathematcs, MIT, Cambrdge, MA Emal: cb lee@math.mt.edu. Research supported by NSF Grant DMS

2 As explaned above, the key tool used n these proofs s the blow-up lemma. A bpartte graph wth parts A B s ε-regular f for every X A and Y B of szes X ε A and Y ε B, the relaton ex,y ) ε holds. Defne the relatve mnmum degree of a bpartte graph wth X Y ea,b) A B parts A B as the mnmum of the two quanttes mn a A da) B db) and mn b B A. In the smplest case for bpartte graphs, the blow-up lemma asserts that for every δ and, there exsts ε such that the followng holds: f G s a bpartte graph wth parts A B for whch A, B) s ε-regular wth relatve mnmum degree at least δ, then t contans all bpartte graphs H wth parts X Y of szes X A and Y B of maxmum degree at most as a subgraph. Another stream of research n the study of embeddng large subgraphs was developed n Ramsey theory. The Ramsey number of a graph H, denoted by rh), s the mnmum nteger n for whch every two-colorng of the edge set of K n, the complete graph on n vertces, contans a monochromatc copy of H. The study of Ramsey numbers was ntated n the semnal paper of Ramsey 3 and now became one of the most actve areas of research n combnatorcs. One mportant problem n Ramsey theory s to understand the asymptotcs of rh) for sparse graphs H. Snce establshng small upper bounds on Ramsey number can be rephrased as fndng large monochromatc subgraphs n edge-colored graphs, t s somewhat nevtable that developments n the two problems are closely related to each other. A graph s d-degenerate f there exsts an orderng of ts vertex set n whch each vertex has at most d neghbors precedng tself. Degeneracy s a natural measure of sparseness snce t mples that every subset of m vertces span at most md edges. The phenomenon of sparse graphs havng small Ramsey number was frst antcpated and studed by Burr and Erdős 6 who made the followng nfluental conjecture and offered a cash prze for a soluton): for every d 1, there exsts a constant c = cd) such that every n-vertex d-degenerate graph H satsfes rh) cd)n. The conjecture s stll open and the current best bound s gven by Fox and Sudakov 16, 17 who, mprovng the result of Kostochka and Sudakov 9, proved that there exsts a constant c d dependng on d such that rh) c d log 1/ n n holds for every d-degenerate n-vertex graph H. Some versons of ths conjecture wth relaxed sparsty condtons have been establshed. The case when H has bounded maxmum degree was solved by Chvátal, Rödl, Szemeréd, and Trotter 9 usng the regularty lemma, and the bound was later mproved n subsequent papers 13, 0, 16, 11. An m-vertex graph H s a-arrangeable f ts vertces can be labelled by m = {1,,, m} so that N N + )) a holds every vertex m where N x) s the set of neghbors of x that have smaller label than x, and N + x) s the set of neghbors of x that have larger label than x). Chen and Schelp 7 ntroduced the concept of arrangeablty n ther study of Burr and Erdős s conjecture and proved that the conjecture holds for graphs wth bounded arrangeablty. Note that arrangeablty s a measure of sparseness of graphs that les strctly between bounded degree and bounded degeneracy. For example, f a graph contans a vertex v n at least 3a+1 trangles sharng no other vertex than v, then t has arrangeablty at least a, but may be -degenerate. Another example s a -subdvson of a complete graph, a graph obtaned from the complete graph by replacng each edge wth nternally vertex-dsjont paths of length.

3 In ths paper, we develop embeddng results of graphs wth bounded degeneracy and small bandwdth. The man theorem s an almost-spannng blow-up lemma for such graphs. For a graph G, a par of vertex subsets X and Y are ε, δ)-dense f for every X X and Y Y of szes X ε X and Y ε Y the number of edges between X and Y s at least δ X Y. An r-partte graph G wth vertex partton V 1 V r s ε, δ)-dense f V, V j ) s ε, δ)-dense for all pars of dstnct ndces, j r. The orgnal blow-up lemma s a spannng embeddng result of bounded degree graphs nto ε, δ)-dense graphs of large enough mnmum degree. Recently, Böttcher, Kohayakawa, Taraz, and Würfl showed that the condton of havng bounded degree can be relaxed nto the condton of havng bounded arrangeablty and maxmum degree at most n log n. The followng theorem further relaxes the condton and asserts that an almost-spannng blow-up lemma holds for degenerate graphs of small bandwdth. Theorem 1.1. For each postve ntegers r, d and postve real numbers ε, ε, δ, satsfyng ε δ )r, there exsts N such that the followng holds for all n N. Let H be an r-partte graph over a vertex partton W ) r that s d-degenerate, has bandwdth at most n 1 ε, and satsfes W 1 ε)n for all r. Let G be an ε, δ)-dense r-partte graph over a vertex partton V ) r where V = n for all r. Then G contans H as a subgraph. See Theorem 4.4 for a slghtly more general form where we do not requre the parts to be of equal sze, and gve a more precse bound on the bandwdth. Note that the complete bpartte graph K d,n d s d-degenerate. In a random bpartte graph of densty 1 wth n vertces n each part, almost surely every d-tuple of vertces have 1 + o1)) n common neghbors and therefore t d does not contan K d,n d as a subgraph. Hence the theorem above does not hold f we completely remove the restrcton on bandwdth. Nevertheless, t mght be possble to replace the restrcton by a moderate restrcton such as the maxmum degree beng at most c n for some constant c. The technque used n provng Theorem 1.1 s dfferent from that used n the prevous versons of the blow-up lemmas. Our embeddng strategy has ts orgn n the methods developed n Ramsey theory. More specfcally, t s based on dependent random choce, a powerful technque n probablstc combnatorcs, and bulds on the deas developed by Kostochka and Sudakov 9, and Fox and Sudakov 16. Ths technque can further be utlzed to extend the bandwdth theorem to almost spannng graphs of bounded degeneracy. An extenson of the bandwdth theorem to arrangeable graphs was proved by Böttcher, Taraz, and Würfl 4 usng the verson of blow-up lemma developed by Böttcher, Kohayakawa, Taraz, and Würfl ther result requres a much more modest bound of on) on the bandwdth of H compared to the n 1 o1) bound of our theorem). Theorem 1.. For all postve ntegers r, d and postve real numbers ε, ε, δ, there exsts N such that the followng holds for all n N. If G s an n-vertex graph of mnmum degree at least 1 1 r + δ)n, and H s an r-partte d-degenerate graph of bandwdth at most n1 ε on at most 1 ε)n vertces, then G contans H as a subgraph. Let G be an n-vertex graph consstng of two clques of order 1 1 r )n sharng 1 r )n vertces. Note that t contans two dsjont sets of vertces each of sze 1 r n whch form an empty bpartte 3

4 graph. Hence r-chromatc graphs wth good expanson property cannot be embedded nto G and thus the bandwdth condton s necessary n Theorem 1.. The mnmum degree condton can be relaxed for bpartte graphs. Theorem 1.3. For all postve ntegers d and postve reals ε, δ, there exsts N such that the followng holds for all n N. If G s an n-vertex graph of mnmum degree at least δn, and H s a bpartte d-degenerate graph of bandwdth at most e 100 d log1/ε) log n n on at most δ ε)n vertces, then G contans H as a subgraph. Snce an n-vertex graph of densty δ contans a subgraph of mnmum degree at least δ n, Theorem 1.3 gves a densty-type embeddng theorem for bpartte degenerate graphs of small bandwdth. Thus n some sense t extends an embeddng result of Fox and Sudakov 16 mprovng that of Kostochka and Sudakov 9), assertng the exstence of a postve constant c such that f H s an d-degenerate bpartte graph on at most e c d log1/δ) log n n vertces, then every n-vertex graph of densty at least δ contans a copy of H. The technque also has applcatons n Ramsey theory. Allen, Brghtwell, and Skokan 1 proved that bounded degree n-vertex r-chromatc graphs of bandwdth on) has Ramsey number at most r +4)n. We combne our technque wth ther framework to prove the followng theorem that brngs us one step closer to the resoluton of Burr and Erdős s conjecture. Theorem 1.4. For all postve ntegers r, d and postve real ε, there exsts N such that the followng holds for all n N. Let H be an n-vertex r-chromatc d-degenerate graph of bandwdth at most n 1 ε. Then rh) r + 5)n. Snce a d-degenerate graph has chromatc number at most d + 1, Theorem 1.4 mples that rh) d + 7)n regardless of the chromatc number of H. Therefore t proves Burr and Erdős s conjecture for graphs of small bandwdth. Furthermore, gven a graph, we can add solate vertces to obtan a graph wth small bandwdth. Hence a more precse form gven n Theorem 5.6 shows that Ramsey numbers of d-degenerate graphs are nearly lnear. Such result was prevously obtaned n 16, 9 wth smlar but better bounds. The phenomenon of degenerate graphs of small bandwdth havng small Ramsey numbers has been observed before n a slghtly dfferent context. Let σh) be the sze of the smallest color class n any proper χh)-colorng of the vertces of H. A smple constructon observed by Chvátal and Harary 8 and strengthened by Burr 5) shows that rg, H) χh) 1) V G) 1) + σh) holds for all graphs H and connected graphs G satsfyng V G) σh). Ramseygoodness, ntated by Burr and Erdős 6, s a topc n Ramsey theory that studes pars of graphs G and H for whch rg, H) equals the rght-hand-sde n the nequalty above see, e.g., 5, 31, 1, 10, 15). Nkforov and Rousseau 31 showed that for each fxed s and d, every suffcently large d-degenerate n-vertex graph G of bandwdth at most n 1 o1) together wth H = K s satsfes the equalty. 1 1 Ther result s stated n terms of separators nstead of bandwdth and n fact s slghtly stronger than what s stated here. 4

5 Note that there s about) a factor of dfference between the Ramsey-goodness bound on rh) and the bound gven n Theorem 1.4 for large values of r. It s lkely that the bound n Theorem 1.4 can be mproved to rh) 1 + c r )rn, where c r s a constant dependng on r that tends to zero as r tends to nfnty see 1, Conjecture 37). We remark that the bandwdth condton plays an mportant role n decdng the constant factor n the ramsey number of sparse graphs. Ths can be seen from a constructon of Graham, Rödl, and Rucńsk 0 establshng that there exsts a constant c such that for every large enough n, there are n-vertex bpartte graphs wth maxmum degree at most havng Ramsey number at least c n. The rest of the paper s organzed as follows. We begn by provdng a bref outlne of the proof of Theorem 1.1 n Secton. Then n Secton 3 we prove Theorem 1.3, and n Secton 4 prove Theorem 1.1. Usng the tools developed n Secton 4, we then proceed to Secton 5 where we prove Theorems 1. and 1.4. We conclude the paper n Secton 6 wth some remarks. Notaton. For an nteger m, defne m := {1,,..., m} and for a par of ntegers m 1, m, defne m 1, m := {m 1,, m }, m 1, m ) := {m 1,, m 1}, m 1, m := {m 1 + 1,, m }, and m 1, m ) := {m 1 + 1,, m 1}. For a set of elements X, we defne X t = X X as the set of all t-tuples n X. Throughout the paper, we wll be usng subscrpts such as n x 1.1 to ndcate that x s the constant comng from Theorem/Lemma/Proposton 1.1. Let G = V, E) be an n-vertex graph. For a vertex v and a set T, we defne degx; T ) as the number of neghbors of x n T, and defne degx) := degx; V ). Defne NT ) := {x : {x, t} E, t T } as the set of common neghbors of vertces n T. For two vertces v, w and a set of vertces T, we defne codegv, w; T ) as the number of common neghbors of v and w n T, and defne codegv, w) := codegv, w; V ). For two sets X and Y, defne EX, Y ) = {x, y) X Y : {x, y} E} and ex, Y ) = EX, Y ). Furthermore, defne ex) = 1 ex, X) as the number of edges n X. Let dx, Y ) = ex,y ) X Y be the densty of edges bewteen X and Y. If G s bpartte wth parts V 1 V, then the relatve degree of a vertex v 1 V 1 s defned as dv 1) V v V s defned as dv ) V 1, and of a vertex. Recall that the relatve mnmum-degree of G s the mnmum of the relatve-degree over all vertces n G. For a graph H, a partal embeddng of H on V V H) s an embeddng of HV nto G. For V V an extenson to V of a partal embeddng f on V s an embeddng g : HV G such that g V = f. We often abuse notaton and denote the extended map usng the same notaton f. Outlne of the Proof We start by descrbng a bref outlne of the proof. Let H be a d-degenerate graph, and let G be a graph to whch we wsh to embed H. Our embeddng strategy s based on an teratve usage of dependent random choce, where each round of teraton s smlar to the one used by Kostochka and Sudakov 9, and by Fox and Sudakov 16. Suppose that H and G are both bpartte graphs wth bpartton W 1 W and V 1 V, respectvely. Kostochka and Sudakov s strategy can be summarzed n the followng steps: 5

6 1. Randomly select s elements n V 1 to fnd a set T V 1 for whch all s+d)-tuples of vertces n A := NT ) have at least V H) common neghbors n V 1.. Randomly select s elements n A to fnd a set S A of sze S s for whch all d-tuples of vertces n A 1 := NS) have at least V H) common neghbors n A. 3. Embed H nto A 1 A. One key observaton s that each d-tuple Q of vertces n A have at least V H) common neghbors n A 1. To see ths, observe that Q S has at least V H) common neghbors n V 1 by Step 1 snce Q S s + d. Snce NQ) A 1 = NQ) NS) V 1 = NQ S) V 1, t mples that Q has at least V H) common neghbors n A 1. Hence n Step 3, one can embed vertces of H one at a tme, followng the d-degenerate orderng of H. Fox and Sudakov s strategy devates from ths n Step 1, where nstead of mposng all s + d)-tuples of vertces n A have enough common neghbors, they bound the number of s + d)-tuples of vertces n A havng nsuffcent common neghbors. Ths then mples that most sets S A of sze S s has the property that for all Q A of sze Q d, the set S Q has enough common neghbors. The man advantage of ths approach s that then we can take s = s; such dfference results n an mprovement on the bound. Thus n Step, we just need to choose such typcal S. The bound on the number of vertces of H comes from the bound on the szes of A 1, A, and the number of common neghbors of d-tuples. Now suppose that H s a larger graph but has small bandwdth, say β. By usng the strategy above, we can embed the ntal segment of H nto A 1 A but wll at some pont run out of space. Say that we have enough space to embed β vertces of H. Start by embeddng the ntal β vertces of H nto A 1 A. The key dea s to halt the embeddng for a moment, and fnd another par of sets B 1 B as above wth the followng addtonal property: *) All d-tuples of vertces n A 1 have at least β common neghbors n A B, and all d-tuples of vertces n A have at least β common neghbors n A 1 B 1. Once we fnd such a par we wll embed the next β vertces nto A 1 B 1 and A B by nvokng ths property. At ths pont, we can remove the mage of the frst β vertces of H from the graph G. Ths s because there are no edges between the ntal segment of β vertces of H and the vertces wth label greater than β. Note that we are left wth a par B 1 B wth β vertces of H embedded nto t. Hence repeatng ths process wll eventually result n an embeddng of H. Thus the man challenge s to fnd a par of sets B 1 B wth the property lsted above. Ths can be done by frst followng the modfed Steps 1 and so that we mpose the addtonal property that most s + d)-tuples n A have many common neghbors n A 1, and vce versa. Then, 4. Randomly select s elements n A 1 to fnd a typcal set T A 1 for whch most s + d)- tuples of vertces n B := NT ) have at least β common neghbors n V Randomly select s elements n A B to fnd a typcal set S A B of sze S s for whch all d-tuples of vertces n B 1 := NS ) have at least β common neghbors n A B. 6

7 Note that snce T s a typcal set, for all d-tuple Q of vertces n A 1, the set Q T has at least β common neghbors n A,.e., Q has at least β common neghbors n A NT ) = A B. Smlarly snce S s a typcal set, all d-tuples n A wll have at least β common neghbors n A 1 B 1, thus establshng *). One very mportant techncal detal needs to be addressed. In Step 5, for our strategy to succeed, t s mportant that A B s consderably larger than β so that a d-tuple of vertces havng less than β common neghbors n A B s an uncommon stuaton. Ths addtonal constrant has a chan effect on our proof. Frst n Step 4, we need to mpose ths addtonal condton of A B beng large when choosng A 1. Second, the sze of A B depends on the number of edges between A 1 and A, and thus n Steps 1 and we must have chosen A 1 and A so that da 1, A ) s large enough. Thrd, to prepare for the next step, when choosng B 1 and B we must also mpose that db 1, B ) s large enough. In the end, t turns out that we need to keep track of certan copes of C 4 or K, ) across varous sets, and ths s where havng G to be an ε, δ)-dense graph becomes necessary. Obtanng such estmates become complcated because our technque dependent random choce) s based on the frst moment method, and we must encode all ths nformaton nto a sngle equaton. For bpartte graphs, there s a way to slghtly modfy Steps 4, 5 and the embeddng strategy to avod usng the ε, δ)-regular condton on G we do not gan n terms of smplcty). The reason we gave a descrpton as above s because t works for non-bpartte graphs H as well. For r-partte graphs, nstead of C 4 we wll be countng copes of complete r-partte graphs wth vertces n each part. Suppose that H s a d-degenerate graph wth bandwdth β. Note that snce the gven graph H s d-degenerate there exsts an orderng n whch all vertces have at most d neghbors that precede tself, and snce H has bandwdth β there exsts an orderng n whch all adjacent pars are at most β apart from each other. In the strategy sketched above, we mplctly used the followng lemma whch asserts that there exsts an orderng of the vertces achevng both. A labellng of an m-vertex graph H by m s d-degenerate f each vertex has at most d neghbors precedng tself, and β-local f all adjacent pars are at most β apart. Lemma.1. Let H be a d-degenerate graph wth bandwdth β. Then there exsts a 5d-degenerate β log 4β)-local labellng of the vertces of H. Proof. Let H be an m-vertex d-degenerate graph wth bandwdth β and denote V = V H). Frst label the vertces by m so that adjacent vertces are at most β apart. Call ths labellng σ : V H) m. We wll construct another labellng π : V H) m whch has the desred property, based on an teratve algorthm where for t = 0, 1,..., m 1, the t-th step of the algorthm defnes the pre-mage π 1 m t). Suppose that we are at the t-th step of the algorthm and let V t = V H) \ π 1 {m t + 1, m t +,..., m}). Choose the vertex v t V t havng at most 5d neghbors precedng tself n V t n the labellng σ, wth maxmum σv t ) note that such vertex exsts snce H s d-degenerate). Defne πv t ) = m t. The defnton mples that n the orderng π, each vertex has at most 5d neghbors precedng tself. We clam that β σv) πv) β log β 7

8 for all v V H). The lemma follows from ths clam snce f {v, w} s an edge, then πv) πw) πv) σv)) + σv) σw)) + σw) πw)) β + β log β + σv) σw)) β log 4β). Fx a value of t and let M t be the set of at most β largest labelled vertces of V t accordng to σ). Note that σm t ) m t β, m. Snce H s d-degenerate, t follows that M t spans at most M t d edges. Also, by the bandwdth condton, the at most) β largest labelled vertces n M t have all ts neghbors n V t nsde M t. Therefore there must exst a vertex among them havng at most M t d β d neghbors precedng tself n σ n V t. Ths mples that σv t ) m t β = πv t ) β, or equvalently, σv t ) πv t ) β. Suppose that σv t ) πv t ) > β log β for some t. Note that there exsts t < t such that σv t ) m t as otherwse {σv 1 ),, σv t 1 )} = m t, m and σv t ) m t = πv t ). Observe that σv t ) m t = πv t ) σv t ) β log β. Note that each w V t wth σw) > σv t ) has at least 5d neghbors precedng tself n σ n V t snce otherwse we would have chosen w nstead of v t. Defne I j = σv t ) jβ, σv t ) j 1)β for j 0, and N 0 = {v t }. Assume that a set N whch s a subset of the -th neghborhood of v t, satsfyng N and N I j for some j s gven. Defne N +1 = {v V t : w N Nv), σv) < σw)}. Note that N +1 N spans less than N +1 + N )d edges by the degeneracy condton. Furthermore f log β, then all vertces n N succeed v t n σ and thus have at least 5d neghbors precedng tself n σ. Therefore N +1 N spans at least N 5d edges. Hence N 5d < N +1 + N )d, and we have N +1 > 4 N. Snce N I j, we have N +1 I j I j +1. Therefore ether N +1 I j or N +1 I j +1 has sze greater than N. Defne N +1 as the ntersecton havng sze greater than N +1. Ths s a contradcton for = log β snce each nterval has sze at most β. Hence the clam σv t ) πv t ) β log β holds. 3 Bpartte graphs Throughout ths secton, fx a bpartte graph H. We wll consder the problem of embeddng H nto another bpartte graph G. We assume that the bpartton W 1 W of H and V 1 V of G are gven. We restrct our attenton to fndng an embeddng f : V H) V G) for whch fw 1 ) V 1 and fw ) V even when we do not explctly refer to the bpartton of H. A set X s d, β)-common nto Y f every d-tuple of vertces of X has at least β common neghbors n Y. A par of sets X, Y ) s d, β)-common f X s d, β)-common nto Y and Y s d, β)-common nto X. Let G be a bpartte graph wth partton V 1 V. The usefulness of d, β)-common property has been notced over years and researchers developed powerful methods such as dependent random choce to fnd subsets that are d, β)-common see, e.g., the survey paper of Fox and Sudakov 18). The followng lemma shows how we wll use the d, β)-common property. 8

9 Lemma 3.1. Let G be a bpartte graph wth bpartton V 1 V, and let H be a bpartte graph wth a d-degenerate β-local vertex labellng by β + β. Suppose that pars of sets A 1, A ) and B 1, B ) wth A 1, B 1 V 1 and A, B V satsfy the followng property: ) A 1 s d, β + β)-common nto A, ) A s d, β + β)-common nto A 1 B 1, and ) A 1 B 1 s d, β + β)-common nto A B. Then every partal embeddng f : β V G) can be extended nto an embeddng f : β + β V G) where all vertces wth label larger than β are mapped nto B 1 B. Proof. We wll extend the partal embeddng one vertex at a tme accordng to the order gven by the vertex labellng. Suppose that the partal embeddng has been defned over t 1 for some t > β, and we are about to embed vertex t. There are at most d neghbors of t that precede t. Call these vertces v 1, v,..., v d for some d d, and let w = fv ) be ther mages. Frst suppose that {w } d =1 A. Then by the gven condton, snce d d, the d -tuple w 1, w,..., w d ) has at least β + β common neghbors n A 1 B 1. Snce H conssts of at most β + β vertces, there exsts at least one vertex x A 1 B 1 that s adjacent to all vertces w 1, w,..., w d and s not an mage of f. By defnng ft) = x, we can extend f to t. Now suppose that {w } d =1 A 1. If d β + β, then the same strategy as above allows us to embed ft) nto B 1. If d > β + β, then note that by the bandwdth condton {v } d =1 β, β + β and thus {w } d =1 A 1 B 1. Then the argument above shows that we can embed ft) nto A B. In the end, we obtan the desred extenson of f. Suppose that a partal embeddng f : β A 1 A s gven. Our proof proceeds by fndng a par of subsets B 1, B ) and applyng Lemma 3.1 to extend f so that the next segment embeds nto B 1 B. Afterwards, we repeat the same process to further extend f untl we fnd an embeddng of the whole graph H. Thus the key step of the proof s to fnd such B 1, B ) gven A 1, A ). We ntroduce one mportant concept before statng the lemma that serves ths purpose. Gven a par of sets X and Y n a graph G, defne the p, d, β)-potental of X n Y as the number of p + d)-tuples n X that have fewer than β common neghbors n Y. We say that X has λ- neglgble p, d, β)-potental n Y f ts p, d, β)-potental n Y s less than λ p 1. We may smply denote the p, d, β)-potental as p-potental when d and β are clear from the context. Note that f X has λ-neglgble 0-potental, then the number of d-tuples n X that have fewer than β common neghbors s less than λ 1 < 1,.e., X s d, β)-common nto Y. Conversely for all p 0, f X s p + d, β)-common nto Y, then X has λ-neglgble p, d, β)-potental n Y. In fact, the reason we have ntroduced p-potental s because t s easer to mpose small p, d, β)-potental than to mpose p + d, β)-commonness. In the bpartte settng, ntroducng ths addtonal concept results n a better bound and the theorem can be proved even wthout the concept ths s how the bound on the Ramsey number of degenerate graphs of Fox and Sudakov 16 dffers from that of Kostochka and Sudakov 9). However for non-bpartte graphs we heavly rely on ths concept 9

10 and thus we ntroduce t here to prepare the reader for the more techncal part that wll later come and for the better bound). Lemma 3.. For natural numbers n, d and postve real numbers ε, γ, δ, defne s = d log n log/δ) and suppose that λ δ 5s ) n, β λ ) 3 n γn, and ε < δ 8. For suffcently large n, let G be a bpartte graph on at most n vertces wth parts V 1 V satsfyng V 1, V γn, where at least 1 ε) V 1 vertces n V 1 have degree at least δ V n V, and vce versa for vertces n V. Suppose that sets A 1 V 1 and A V for whch A contans at least 1 δ ) s 4 V vertces of degree at least δ V 1 s gven. Then there exst sets B 1 V 1 and B V satsfyng the followng propertes: ) B 1 s d, β)-common nto A B, ) B has λ-neglgble s, d, β)-potental n B 1, and ) B contans at least 1 δ ) s V vertces of degree at least δ V 1. Furthermore, f A has λ-neglgble s, d, β)-potental n A 1, then v) A s d, β)-common nto A 1 B 1, The proof of Lemma 3. wll be gven n the next subsecton. We frst prove the man result of ths secton usng Lemmas 3.1 and 3.. A bpartte graph G wth bpartton V 1, V ) s α, ε, δ)- degree-dense f for every par of subsets X 1, X ) satsfyng X V and X α V for = 1, ), the subgraph G nduced on X 1 X contans at least 1 ε) X vertces n X of relatve degree at least δ for = 1, ). Note that f G s ε, δ)-dense, then t s ε 1, ε, δ)-degree-dense for all ε 1 ε ε. Also, f G has relatve mnmum degree α, then t s 1 α + δ, 0, δ)-degree-dense for all δ < α. Hence Theorem 1.3 and the bpartte case of Theorem 1.1 follows from the followng theorem. As mentoned before, the technques used n ths secton s talored to bpartte graphs. As a result, the bound on the bandwdth that we obtan here for the bpartte case of Theorem 1.1 s n fact better than the bound that we obtan through the proof of the general case of Theorem 1.1 that wll be gven n a later secton. Theorem 3.3. Let ε, δ, α, γ be fxed real numbers satsfyng 0 ε < δ 8. Let H be a d 0-degenerate bpartte graph wth parts W 1 W havng bandwdth β 0 e 50 log1/δ)d 0 log n γn. If n s suffcently large and G s an n-vertex α, ε, δ)-degree-dense bpartte graph wth bpartton V 1 V satsfyng V max{γn, 1 1 α W } for = 1,, then G contans a copy of H. Proof. Let H be a gven m-vertex d 0 -degenerate bpartte graph of bandwdth β 0 wth bpartton W 1 W. For d = 5d 0 and β = β 0 log 4β 0 ), by Lemma.1 there exsts a d-degenerate β-local labellng of the vertex set of H. Defne I = 1)β, β m and J = β m for 0. For < 0, defne I = J =. Defne s = d log n log/δ) and λ = δ )5s n. Note that β 1 4 λ n )3 αγn. Let G be a gven n-vertex bpartte graph wth bpartton V 1 V, where W 1 1 α) V 1 and W 1 α) V. Let V = V G) and n = V. Our goal s to fnd an embeddng f of H to G for whch fw ) V 10

11 for = 1,. We wll embed H usng an teratve algorthm. For t 1, at the begnnng of the t-th step we wll be gven a par A 1, A ) wth A 1 V 1 and A V and a partal embeddng f : J t 1 V G) satsfyng the followng propertes: ) fi t 1 ) A 1 A V \ fj t ), ) A 1 s d, 4β)-common nto A, ) A has λ-neglgble s, d, 4β)-potental n A 1, v) A contans at least 1 δ ) s 4 V \fj t ) vertces of relatve-degree at least δ n V 1 \fj t ). Then at the t-th step we wll construct sets C 1 V 1 and C V, and extend f to I t so that the condtons above for the t + 1)-th step s satsfed. We can then contnue the process untl we fnsh embeddng H. Intally, apply Lemma 3. to the sets A 1 ) 3. = V 1, A ) 3. = V, and β 3. = β to obtan sets A 0) 1 and A 0) satsfyng Propertes ) and ). Note that Propertes ) and v) are vacuously true snce I = J = for < 0. Suppose that for some t 1, we completed the t 1)-th step of the algorthm. Let G t be the subgraph of G nduced on V \ fj t ). Snce G s α, ε, δ)-degreedense and W 1 α) V for = 1,, t follows that at least 1 ε proporton of vertces of each part of the bpartton of G t has relatve degree at least δ. Note that V V G t ) α V αγn for = 1,. By Propertes ) and v) we may apply Lemma 3. to the par A 1, A ) n the graph G t wth β 3. = 4β and γ 3. = αγ to obtan sets B 1 and B satsfyng the followng propertes: a) B 1 s d, 8β)-common nto A B, b) B has λ-neglgble s, d, 8β)-potental n B 1, c) B contans at least 1 δ ) s V \ fj t ) vertces of degree at least δ V 1 \ fj t ), and d) A s d, 4β)-common nto A 1 B 1, Note that fi t 1 ) V G t ). Let g = f It 1. Snce g s an embeddng of HI t 1 nto G t, t can also be vewed as a partal embeddng of HI t 1 I t nto G t defned on I t 1. Thus by Property ) and Propertes a), d), we can apply Lemma 3.1 and extend g to I t so that gi t ) A 1 B 1 ) A B ) B 1 B. Furthermore, snce the labellng s β-local, there are no edges of H between I t and J t. Thus g n fact extends f to I t. Defne C 1 = B 1 \ fi t 1 ) and C = B \ fi t 1 ). The sets fi t 1 ) and fi t ) are dsjont snce f s a partal embeddng. Thus fi t ) C 1 C, and Property ) for the next step s satsfed. Defne G t+1 as the subgraph of G nduced on V \ fj t 1 ). Note that G t+1 s obtaned from G t by removng β vertces. Property ) for the next step then follows from Property a). By Property b), B has λ-neglgble s, d, 8β)-potental n B 1 n the graph G t. An s + d)-tuple n B can have less than 4β common neghbors n C 1 only f t has less than 6β common neghbors n B 1. Thus B has λ-neglgble s, d, 6β)-potental n C 1, and snce C B, t follows that C has λ-neglgble s, d, 6β)-potental n C 1, provng Property ) for the next step. Furthermore by Property c), there are at least 1 δ ) s V \fj t ) vertces n B of degree at least δ V 1 \fj t ) n the graph G t. Therefore C contans at least 1 δ ) s V \ fj t ) β 1 δ ) s 4 V \ fj t 1 ) 11

12 vertces of degree at least δ V 1 \ fj t ) β δ V 1 \ fj t 1 ) n the graph G t+1. Ths proves Property v) for the next step and concludes the proof. Theorem 1.3 straghtforwardly follows. Proof of Theorem 1.3. Let G be an n-vertex graph of mnmum degree at least γn, and let V = V G). Let H be a d-degenerate bpartte graph wth parts W 1 W havng bandwdth at most e 100 d log1/ε) log n n where W 1 + W γ ε)n. Defne w 1 = W 1 + ε 4 n and w = W + ε 4 n. Let V 1 V be a bpartton of V wth V 1 = w 1 w 1 +w n and V = w w 1 +w n chosen unformly at random. Standard esmates on concentraton of hypergeometrc nequalty shows that the bpartte subgraph of G nduced on V 1 V has relatve mnmum degree at least γ ε 4 wth probablty 1 o1). Let V 1 V be a partcular partton where such bound holds and let G be the bpartte subgraph nduced on V 1 V. Snce G has relatve mnmum degree at least γ ε 4, t s 1 γ + ε, 0, ε 4 )-degree-dense. Note that mn{ V 1, V } ε 4n. Furthermore, V 1 = w 1 n W 1 + ε/4)n W 1 w 1 + w γ ε/) γ ε/), and smlar bound holds for V. Therefore we can apply Theorem 3.3 wth ε) 3.3 = 0, δ) 3.3 = ε 4, α) 3.3 = 1 γ + ε, and γ) 3.3 = ε 4 to fnd a copy of H n G. 3.1 Proof of Lemma 3. In ths subsecton we prove Lemma 3. usng dependent random choce. We wll take B = NT 1 ) for some approprately chosen random set T 1 V 1. The man challenge s that the events that we would lke to control are not concentrated more precsely, we do not know how to prove that they are concentrated), and hence we need to encode all the events nto a sngle random varable and use the frst moment method. For example f we want to show that B b, then we can show E B b 0. Then there exsts a choce of T 1 for whch B b. Our stuaton s more complcated. We want to fnd B so that B b and B A a smultaneously holds among other propertes). We can try to prove that E B B A ab 0, but then we have no bound on ndvdual sets. Ths s crtcal for us snce we want a lower bound on B that s ndependent of the sze of A otherwse the set B wll rapdly shrnk between teratons and our embeddng strategy wll fal). What we can do nstead s prove that E B B A a B b A B 0. As long as a and b are postve real numbers, ths wll mply the desred bounds. In fact we do not need an ndvdualzed bound on A B and hence our random varable wll be of the form E B B A a B ab 0, 1

13 whch s easer to verfy. We also need bounds on certan potentals. For two sets X and Y, denote the p, d, β)-potental of X n Y as ξ p,d,β X, Y ). If we want to prove that the p, d, β)-potental of B n V 1 s λ-neglgble,.e., ξ p,d,β B, V 1 ) < λ p 1, then we wll modfy the equaton nto E B B A a B ab n ξ p,d,β B, V 1 ) λ p 1 > 0, as t wll force ξ p,d,β B, V 1 ) < B B A λ p 1 λ p 1. In some cases we can only bound the n potental n terms of another potental and need control on the rato between two potentals. The followng proposton captures the strength and beauty of dependent random choce. Proposton 3.4. Let s, d, p be ntegers. Let X and Y be a gven par of subsets of vertces and let X X be a subset of sze at least m. Let ˆT be a s-tuple of vertces n X chosen ndependently and unformly at random. Then the followng hold. ) ) E E ξ p,d,β N ˆT ) ) Y, X) β s. ξ p,d,β Y, X) m ξ p,d,β X, N ˆT ) ) Y ) 1 s. ξ p+s,d,β X, Y ) m Proof. ) Let Q be a fxed p + d)-tuple n Y wth less than β common neghbors n X. The probablty that Q N ˆT ) s NQ) X ) s ) β s X. m Therefore and Part ) follows. E ξ p,d,β N ˆT ) β s ) Y, X) ξ p,d,βy, X), m ) Note that a p + d)-tuple Q n X has less than β common neghbors n N ˆT ) Y f and only f the p + d + s)-tuple Q ˆT has less than β common neghbors n Y. Therefore, Eξ p,d,β X, N ˆT ) Y ) = qξ p+s,d,β X; Y ), where q s the probablty that ˆT s precsely the last s elements of Q for a fxed p + d + s)-tuple Q n X. Part ) follows snce q = 1 m s. We now prove Lemma 3.. Let G be a bpartte graph wth bpartton V 1 V where at least 1 ε proporton of vertces of each sde have relatve-degree at least δ. Suppose that sets A 1 V 1 and A A V are gven where A 1 δ ) s 4 V and all vertces n A have degree at least 13

14 δ V 1 n V 1. Let V 1 V 1 and V V be the set of vertces of relatve-degree at least δ. The gven condton mples that V 1 1 ε) V 1 and V 1 ε) V. Throughout the proof we wll be consderng p, d, β)-potentals for fxed d and two dfferent values of β. Hence to avod havng three subscrpts such as n ξ p,d,β, we abuse notaton and for sets X and Y and an nteger p, we denote the the p, d, β)-potental of X n Y as ξ p X, Y ) and the p, d, β)-potental of X n Y as η p X, Y ). Let ˆT 1 be a random mult-set of s vertces n V 1, where each vertex s chosen unformly and ndependently at random. Set ˆB = N ˆT 1 ) and ˆB = N ˆT 1 ) V. Defne ) µ := E ˆB A ˆB δ s V ˆB A n λ s 1 η s ˆB, V 1 ), and let T 1 be a partcular choce of ˆT1 for whch the random varable on the rght-hand sde becomes at least ts expected value. Let B = ˆB and B = ˆB for ths choce of ˆT 1. Let ˆT be a random mult-set of s vertces n A B, where each vertex s chosen unformly and ndependently at random. Set ˆB 1 = N ˆT ) and defne ν := E η 0 ˆB 1, A B ) + λs η s B, ˆB 1 ) η s B, V 1 ) + λs ξ 0 A, ˆB 1 A 1 ), ξ s A, A 1 ) and let T be a partcular choce of ˆT for whch the random varable on the rght-hand sde becomes at most ts expected value. Let B 1 = ˆB 1 for ths choce of ˆT see Fgure 1). V 1 A 1 A V T T 1 B 1 = NT ) B = NT 1) Fgure 1: The sets V, A, B, T for = 1,. We frst choose T 1, and then T. Clam 3.5. µ 1 8 δ ) 4s V and ν < 1. 14

15 We frst deduce the lemma provded that the clam holds. Property ) follows snce B A B ) δ s V B A µ > 0, and V 1 ε) V mples B δ )s V 1 ε) δ )s V. Furthermore snce n λ s 1 η sb, V 1 ) < B A B n = η s B, V 1 ) < λ s 1. 1) Snce ν < 1, we see that η 0 B 1, A B ) + λs η s B, B 1 ) η s B, V 1 ) + λs ξ 0 A, B 1 A 1 ) < 1. ξ s A, A 1 ) The frst term on the left-hand sde gves η 0 B 1, A B ) < 1. Thus B 1 s d, β)-common nto A B and Property ) follows. The second term gves η s B, B 1 ) < λ s η s B, V 1 ) < λ s 1 from 1)), establshng that B has λ-neglgble s, d, β)-potental nto B 1, whch s Property ). The thrd term gves ξ 0 A, B 1 A 1 ) < λ s ξ s A, A 1 ). Hence f A has λ-neglgble s, d, β)-potental nto A 1, then ξ s A, A 1 ) < λ s 1 and thus ξ 0 A, B 1 A 1 ) = 0. Therefore Property v) follows. It thus suffces to prove Clam 3.5. Proof of Clam 3.5. Recall that µ conssts of three terms. For the second term, by lnearty of expectaton E ˆB A = P x N ˆT ) 1 ) = degx; V 1 ) ) s V 1. ) Therefore by convexty, E ˆB A x A A 1 A x A degx; V V 1 x A 1 ) s ) δ s ε A, 3) where the second nequalty follows snce all vertces n A have degree at least δ V 1 and V 1 1 ε) V 1. For the frst term of µ, by lnearty of expectaton, E ˆB A ˆB = x A = x V x A x V P V x A x, x N ˆT 1 ) codegx, x ; V V 1 1 V x V ) 1 ) ) s codegx, x ; V V 1 1 ) s, 4) 15

16 where the fnal nequalty follows from convexty. For fxed x A, the sum x V codegx, x ; V 1 ) counts the number of paths of length startng at x whose second vertex s n V 1 and thrd n V. Therefore by the mnmum degree condton we have codegx, x ; V 1) degx; V 1) δ V ε V ). Hence from 4) and ), x V E ˆB A ˆB x A 1 ) degx; V V δ ε) s V By Proposton 3.4 ), ) n E λ s 1 η s ˆB, V 1 ) n β s λ s 1 η sv, V 1 ) V 1. 1 ) s = δ ε) s V E ˆB A. 5) Observe that η s V, V 1 ) < n s+d trvally holds snce η s V, V 1 ) counts the number of s + d)- tuples n V wth less than β common neghbors n V 1. Recall that s = d log n log/δ), λ δ 5s ) n, β λ n ) 3 γn, and V 1 1 ε)γn. Hence n E λ s 1 η s ˆB, V 1 ) ns+d+ β λ s 1 V λnd+ 1 ε) s Therefore from 5), 3) and ε < δ 8, we obtan µ = E ˆB A ˆB 1 λ n ) δ s V 1 δ ε)s V E ˆB A ) δ s V A 1 ) δ 4s V. 8 ) s λnd+ n 1 ε) s γ s λ β n ) s = λnd+ δ 1 ε) s ) s ) 5s = λn 4d+ 1 ε) s < 1. ˆB A n λ s 1 η s ˆB, V 1 ) We now compute the expected value of ν. The choce of B n partcular mples that B A B 1 8 and snce B V, t follows that B A 1 8 ν = E ) δ 4s V, δ ) 4s V. Recall that η 0 ˆB 1, A B ) + λs η s B, ˆB 1 ) η s B, V 1 ) + λs ξ 0 A, ˆB 1 A 1 ) ξ s A, A 1 ) 16.

17 By Proposton 3.4 ), snce β λ 3 n) γn δ 15s ) γn, E η 0 ˆB β 1, A B ) η 0 V 1, A B ) B A ) < n d β s δ B A n d ) s ) 10s = n 9d < 1. By Proposton 3.4 ), λ s ξ 0 A, E ˆB 1 A 1 ) + η sb, ˆB 1 ) ξ s A, A 1 ) η s B, V 1 ) and thus ν < 1. λ B A ) s < 1, The notatons V 1, V, A, B used n ths subsecton are somewhat cumbersome but plays an mportant role n the proof. Recall that µ = E ˆB A ˆB ) δ s V ˆB A n λ s η s ˆB, V 1 ). The second term ) δ s V ˆB A s extremely mportant snce t allows us to gve a lower bound on the sze of B that s ndependent of the sze of A. The reason we were able to conclude that µ s postve even wth ths term s 5) whch were deduced from the fact that we are workng wth vertces n V, not V. The same phenomenon shows up n greater depth for non-bpartte graphs. Thus t s useful to dentfy these sets of vertces havng large degree. It may seem lke one can avod usng these notatons. One way s by defnng B as some subset of NT 1 ) nstead of B = NT 1 )). However such approach fals snce f B NT 1 ) then T 1 Q havng many common neghbors no longer mples that Q has many common neghbors n B we crucally reled on such property throughout the proof). Another way s by mposng the condton that all vertces n A have relatve degree at least δ n A, and prove that all vertces n B have relatve degree at least δ n A as well. The problem wth ths approach s that we embed β vertces of H after constructng the sets A 1 and A. Once we embed these vertces, the set A has relatve degree at least δ n the remanng set for some δ < δ. Snce B necessarly ntersects A, ths mples that B can only be guaranteed to have relatve degree at least δ nstead of δ). Moreover, after embeddng another β vertces, t wll drop to relatve degree at least δ for some δ < δ. Therefore the embeddng algorthm becomes unsustanable under ths approach. 4 General graphs Throughout ths secton, fx an r-partte graph H. We wll consder the problem of embeddng H nto another r-partte graph G. We assume that r-parttons W ) r of H and V ) r of G are gven. We restrct our attenton to fndng an embeddng f : V H) V G) for whch fw ) V for all r even when we do not explctly refer to the r-partton of H. 17

18 The embeddng lemma that we use for r-partte graphs s n fact smpler than the one used n the prevous secton Lemma 3.1). Such smplcty s acheved at the cost of havng further restrcton on the gven r-tuple A ) r. Throughout ths secton, for an r-tuple of sets A = A ) r, we use the notaton A to denote the set A := j r,j A j. We say that A s d, β)- common f A s d, β)-common nto A for every r. We say that an r-tuple T = T ) r s d, β)-typcal for A f for every r and d-tuple of vertces Q n A, the set Q T has at least β common neghbors n A, or equvalently, Q has at least β common neghbors n A NT ). Note that f T ) r s d, β)-typcal for A ) r, then A s d, β)-common nto A NT ) for all r. Lemma 4.1. Let G be an r-partte graph wth vertex partton A ) r. Let H be an r-partte graph wth vertex partton W ) r and a d-degenerate labellng by β +β. Suppose that an r-tuple of sets T ) r s d, β +β)-typcal for A ) r. Then every partal embeddng f : β r A can be extended to an embeddng f : β + β r A where all vertces wth label greater than β are mapped nto r A NT )). Proof. As mentoned above, the gven condton mples that A s d, β + β)-common nto A := A NT ). We wll extend the partal embeddng f one vertex at a tme accordng to the order gven by the vertex labellng. Suppose that for some t > β, the partal embeddng has been defned over t 1, and we are about to embed vertex t W j for some ndex j r. There are at most d neghbors of t that precede t. Call these vertces v 1, v,..., v d for d d), and let w = fv ) be ther mages. Note that {w } d A j. Snce A j s d, β + β)-common nto A j, the vertces {w } d have at least β + β common neghbors n A j. Snce H conssts of β + β vertces, there exsts at least one vertex x A j sadjacent to all vertces w 1, w,..., w d and s not an mage of f. By defnng ft) = x, we can extend f to t. We can fnd the desred extenson f by repeatng ths process. In order to fnd sets that satsfy the condtons of Lemma 4.1, we need to mantan control on potentals of sets and the number of copes of K r across gven famly of sets. For an r-tuple of sets A = A ) r, we say that a copy of K r s across A f t has one vertex n each set A for r. Let G be an r-partte graph wth vertex partton V ) r. A copy of K r across V ) r wth vertex set v ) r s δ-heavy f for each j r, the r 1)-tuple of vertces v ) r\{j} has at least δ V j common neghbors n V j. Defne κ δ A) as the number of δ-heavy copes of K r across A. Sometmes we addtonally denote the graph G such as n κ δ A; G) to specfy that we are countng copes of K r n G. Throughout the proof, we carefully keep track of δ-heavy copes of K r. Ths corresponds to the fact that we were keepng track of the vertces of large degree n the prevous secton. Recall that for a par of sets X and Y, the p, d, β)-potental of X n Y s the number of p + d)-tuples n X that have fewer than β common neghbors n Y. Further recall that X has λ-neglgble p, d, β)-potental n Y f ts p, d, β)-potental n Y s less than λ p 1. For an r-tuple of sets A = A ) r, we say that A has λ-neglgble p, d, β)-potental f A has λ-neglgble p, d, β)-potental n A for all r. The followng smple fact about potentals wll be useful. 18

19 Proposton 4.. If X has λ-neglgble p, d, β)-potental n Y for some λ X, then t has λ-neglgble p, d, β)-potental n Y for all p p. Proof. Let T be a p -tuple of vertces n X wth less than β common neghbors n Y. Then for every p p )-tuple T, the p-tuple T T has less than β common neghbors n Y. Therefore ξ p,d,βx, Y ) X p p ξ p,d,β X, Y ) < λ p 1. Snce X λ, t follows that ξ p,d,βx, Y ) < λ p 1. The next lemma, correspondng to Lemma 3. n the bpartte case, produces sets satsfyng the condton of Lemma 4.1. As n the prevous secton, we defer ts proof nto another subsecton. Lemma 4.3. Let ε, δ, r be fxed parameters and n be a suffcently large nteger. Let G be an r-partte ε, δ)-dense graph wth vertex partton V ) r, and A = A ) r be an r-tuples of sets wth A V for all r. For s = d log n 1/r), log log n) suppose that the followng condtons hold, a) ε ) δ r, λ n ) ) n 0 5r 10s) r 1 δ n log n, and β n λ r, n) b) n 0 V n for all r, c) κ δ/) ra) 1 δ log n) r 10s)r r =1 V. Then there exsts an r-tuple T = T ) r satsfyng the followng propertes: ) B := V NT )) r s d, β)-common, ) B has λ-neglgble r 1)s, d, β)-potental, and ) κ δ rb) δ log n) r 10s)r r =1 V. Furthermore, f A has λ-neglgble r 1)s, d, β)-potental, then v) T s d, β)-typcal for A. We now prove Theorem 1.1 n a slghtly more general form as stated below, usng the tools developed n ths subsecton. Theorem 4.4. For fxed ε, δ, r satsfyng ε δ )r, there exsts c such that the followng holds. Suppose that β 0 e cd 0 log n) r 1)/r log log n) 1/r γ 4r n and n s suffcently large. Let H be a d 0 - degenerate graph wth bandwdth β 0 and an r-partton W ) r. If G s an n-vertex ε, δ)-dense r-partte graph wth vertex partton V ) r satsfyng V max{ 1 1 ε W, γn} for all r, then G contans a copy of H. ) Proof. Defne s = d log n 1/r, log log n λ = δ log n) 5r 10s)r 1 ε γ n, and n 0 = εγn. Let H be a gven m-vertex d 0 -degenerate r-partte graph of bandwdth β 0 wth vertex partton W ) r. By Lemma.1, there exsts a labellng of the vertex set of H by m n whch all adjacent pars 19

20 are at most β := β 0 log 4β 0 ) apart and all vertces have at most d := 5d 0 neghbors precedng tself. For suffcently large c dependng on ε, δ, r, the gven bound on β 0 mples β n λ r n) for large enough n. Let G be a gven n-vertex r-partte graph on V := V G) wth vertex partton V ) r, where W 1 ε) V for all r. Our goal s to fnd an embeddng f of H nto G for whch fw ) V for all r. Defne I t = t 1)β, tβ m for t 1. We wll embed H usng an teratve algorthm. For t 1, at the begnnng of the t-th step, we are gven an r-tuple of sets A = A ) r and a partal embeddng f : t 1)β V H) satsfyng the followng propertes. Defne V t) := V \ ft )β) for all r. ) A V t) for all r, ) fi t 1 ) r A. ) A has λ-neglgble r 1)s, d, β)-potental, and ) r v) κ δ/) ra; G t) ) 1 10s) r δ r t) log n =1 V where G t) s the subgraph of G nduced on r V t). Gven A and f as above, the t-th step of the teraton fnds an r-tuple of sets B = B ) r and extends f so that the condtons for the t + 1)-th step s satsfed where A s replaced wth B). The process can then be contnued untl we fnd an embeddng of H nto G. As an ntal step, apply Lemma 4.3 to A 4.3 = V ) r to fnd an r-tuple of sets A 0) = A 0) that s d, β)-common, has λ-neglgble r 1)s, d, 4β)-potental, and satsfes κ δ ra 0) ) δ log n) r 10s)r r =1 V. Ths provdes the r-tuple needed to begn the step t = 1 where f s the partal embeddng defned on the empty-set). Now suppose that for some t 1, we completed the t 1)-th step of the algorthm and are gven A and f satsfyng the propertes lsted above. Snce G s ε, δ)-dense and W 1 ε) V for all r, the subgraph G t) s ε, δ)-dense. Thus we can apply Lemma 4.3 to the graph G t) wth n 0 ) 4.3 = εn 0, β 4.3 = β, and A 4.3 = A, and obtan an r-tuple T = T ) r for whch C := NT ) V and C := C ) r satsfes the followng propertes: ) C s d, 4β)-common, ) C has λ-neglgble r 1)s, d, 4β)-potental, ) r ) κ δ rc; G t) 10s) r ) δ r t) log n =1 V, and v ) T s d, β)-typcal for A. Consder f It 1 as a partal embeddng of HI t 1 I t defned on I t 1 and apply Lemma 4.1 to extend t to I t. Snce H has no edge between I t and t )β, t gves an extenson of f to I t so that fi t ) r A C ) r C. Defne B := C \ fi t 1 ) for each r and denote B = B ) r. Snce C V t), t follows that B V t+1) for all r. Thus B satsfes Property ) for the t-th step. It also satsfes Property ) snce fi t ) and fi t 1 ) are dsjont by the defnton of embeddng and fi t ) r C. 0 ) r

21 Note that G t+1) s obtaned from G t) by removng β vertces. Hence for a set of vertces X, ts p, d, β)-potental n B s at most ts p, d, 3β)-potental n C, whch s at most ts p, d, 4β)-potental n C, for all r. Thus Property ) follows from Property ). Snce δ r εn 0 β δ )r εn 0, every δ r -heavy copy of K r n G t) s δ )r -heavy n G t+1). Therefore κ δ/) rb; G t+1) ) κ δ rb; G t) ) β n r =1 whch verfes Property v) and concludes the proof. 4.1 Proof of Lemma 4.3 V t) 1 ) δ r 10s) r r V t+1), log n =1 In ths subsecton, we prove Lemma 4.3 through a slghtly more general lemma applcable to later applcatons as well Generalzed verson For an r-tuple of sets A = A ) r, we say that an r-partte graph F s across A, or crosses A, f the -th part of F s n A for all r. For a famly F of r-partte subgraphs of G, we use the notaton FA) to denote the subfamly of graphs n F that cross A. For two graphs F 1 and F that cross A, we say that F 1 and F are adjacent f for every dstnct j, j r, all vertces n the j-th part of F 1 s adjacent to all the vertces of the j -th part of F. For a famly F of graphs that cross A, we say that a graph F s δ-heavy wth respect to F f there are at least δ F graphs n F adjacent to F. For another famly of graphs F, we say that F s δ-heavy wth respect to F f each graph n F s δ-heavy wth respect to F. Recall that a copy K of K r across an r-partton V ) r whose vertces are gven by v ) r s δ-heavy f for each j r, the vertces v ) r\{j} have at least δ V common neghbors n V. Thus f K s δ-heavy, then t s δ r -heavy wth respect to the famly of all copes of the empty r-vertex graph across the partton. The followng lemma mples Lemma 4.3. Lemma 4.5. Suppose that parameters satsfyng the followng condtons are gven: ) { β 1/c+r+1) λ n0 ) ) n n mn δ 10s) r 1 ) } 1 1/s n log, n rn d+r+f+1, where n s suffcently large. Let G be an r-partte graph wth vertex partton V ) r satsfyng n 0 V n for all r. Let F 0 be a f-vertex r-partte graph and F be a famly of copes of F 0 across V ) r. Let K be a famly of copes of K r that s δ 1 -heavy wth respect to F. Suppose that an r-tuple of sets A = A ) r wth A V satsfy KA) δ r =1 V. Then there exsts an r-tuple T = T ) r satsfyng the followng propertes: ) B := V NT )) r has λ-neglgble s, d, β)-potental for all s cs, and ) FB) log n ) 10s) r F. 1