HYPERGRAPH F -DESIGNS FOR ARBITRARY F

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1 HYPERGRAPH F -DESIGNS FOR ARBITRARY F STEFAN GLOCK, DANIELA KÜHN, ALLAN LO AND DERYK OSTHUS Abstract. We solve the exstence problem for F -desgns for arbtrary r-unform hypergraphs F. In partcular, ths shows that, gven any r-unform hypergraph F, the trvally necessary dvsblty condtons are suffcent to guarantee a decomposton of any suffcently large complete r-unform hypergraph G = K n (r) nto edge-dsjont copes of F, whch answers a queston asked e.g. by Keevash. The graph case r = 2 forms one of the cornerstones of desgn theory and was proved by Wlson n The case when F s complete corresponds to the exstence of block desgns, a problem gong back to the 19th century, whch was frst settled by Keevash. More generally, our results extend to F -desgns of quas-random hypergraphs G and of hypergraphs G of sutably large mnmum degree. Our approach bulds on results and methods we recently ntroduced n our new proof of the exstence conjecture for block desgns. 1. Introducton 1.1. Background. A hypergraph G s a par (V, E), where V = V (G) s the vertex set of G and the edge set E s a set of subsets of V. We often dentfy G wth E, n partcular, we let G := E. We say that G s an r-graph f every edge has sze r. We let K n (r) denote the complete r-graph on n vertces. Let G and F be r-graphs. An F -decomposton of G s a collecton F of copes of F n G such that every edge of G s contaned n exactly one of these copes. (Throughout the paper, we always assume that F s non-empty wthout mentonng ths explctly.) More generally, an (F, λ)-desgn of G s a collecton F of dstnct copes of F n G such that every edge of G s contaned n exactly λ of these copes. Such a desgn can only exst f G satsfes certan dvsblty condtons (e.g. f F s a graph trangle and λ = 1, then G must have even vertex degrees and the number of edges must be a multple of three). If F s complete, such desgns are also referred to as block desgns. The queston of the exstence of such desgns goes back to the 19th century. The frst general result was due to Krkman [17], who proved the exstence of Stener trple systems (.e. trangle decompostons of complete graphs) under the approprate dvsblty condtons. In a groundbreakng seres of papers whch transformed the area, Wlson [32, 33, 34, 35] solved the exstence problem n the graph settng (.e. when r = 2) by showng that the trvally necessary dvsblty condtons mply the exstence of (F, λ)-desgns n K n (2) for suffcently large n. More generally, the exstence conjecture postulated that the necessary dvsblty condtons are also suffcent to ensure the exstence of block desgns wth gven parameters n K n (r). Answerng a queston of Erdős and Hanan [10], Rödl [27] was able to gve an approxmate soluton to the exstence conjecture by constructng near optmal packngs of edge-dsjont copes of K (r) f n K n (r),.e. packngs whch cover almost all the edges of K n (r). (For ths, he ntroduced hs now famous Rödl nbble method, whch has snce had a major mpact n many areas.) More recently, Kuperberg, Lovett and Peled [22] were able to prove probablstcally the exstence of non-trval desgns for a large range of parameters (but ther result requres that λ s comparatvely large). Apart from ths, progress for r 3 was manly lmted to explct constructons for rather restrctve parameters (see e.g. [7, 31]). Date: 8th June The research leadng to these results was partally supported by the EPSRC, grant nos. EP/N019504/1 (D. Kühn) and EP/P002420/1 (A. Lo), by the Royal Socety and the Wolfson Foundaton (D. Kühn) as well as by the European Research Councl under the European Unon s Seventh Framework Programme (FP/ ) / ERC Grant Agreement no (S. Glock and D. Osthus). 1

2 2 S. GLOCK, D. KÜHN, A. LO AND D. OSTHUS In a recent breakthrough, Keevash [15] proved the exstence of (K (r) f, λ)-desgns n K(r) n for arbtrary (but fxed) r, f and λ, provded n s suffcently large. In partcular, hs result mples the exstence of Stener systems for any admssble range of parameters as long as n s suffcently large compared to f (Stener systems are block desgns wth λ = 1). The approach n [15] nvolved randomsed algebrac constructons and yelded a far-reachng generalsaton to block desgns n quasrandom r-graphs. Ths n turn was extended n [12], where we developed a nonalgebrac approach based on teratve absorpton, whch addtonally yelded reslence versons and the exstence of block desgns n hypergraphs of large mnmum degree. Ths naturally rases the queston of whether F -desgns also exst for arbtrary r-graphs F. Here, we answer ths affrmatvely, by buldng on methods and results from [12] F -desgns n quasrandom hypergraphs. We now descrbe the degree condtons whch are trvally necessary for the exstence of an F -desgn n an r-graph G. For a set S V (G) wth 0 S r, the (r S )-graph G(S) has vertex set V (G) \ S and contans all (r S )-subsets of V (G) \ S that together wth S form an edge n G. (G(S) s often called the lnk graph of S.) Let δ(g) and (G) denote the mnmum and maxmum (r 1)-degree of an r-graph G, respectvely, that s, the mnmum/maxmum value of G(S) over all S V (G) of sze r 1. For a (non-empty) r-graph F, we defne the dvsblty vector of F as Deg(F ) := (d 0,..., d r 1 ) N r, where d := gcd{ F (S) : S ( ) V (F ) }, and we set Deg(F ) := d for 0 r 1. Note that d 0 = F. So f F s the Fano plane, we have Deg(F ) = (7, 3, 1). Gven r-graphs F and G, G s called (F, λ)-dvsble f Deg(F ) λ G(S) for all 0 r 1 and all S ( ) V (G). Note that G must be (F, λ)-dvsble n order to admt an (F, λ)-desgn. For smplcty, we say that G s F -dvsble f G s (F, 1)-dvsble. Thus F -dvsblty of G s necessary for the exstence of an F -decomposton of G. As a specal case, the followng result mples that (F, λ)-dvsblty s suffcent to guarantee the exstence of an (F, λ)-desgn when G s complete and λ s not too large. Ths answers a queston asked e.g. by Keevash [15]. In fact, rather than requrng G to be complete, t suffces that G s quasrandom n the followng sense. An r-graph G on n vertces s called (c, h, p)-typcal f for any set A of (r 1)- subsets of V (G) wth A h we have S A G(S) = (1 ± c)p A n. Note that ths s what one would expect n a random r-graph wth edge probablty p. Theorem 1.1 (F -desgns n typcal hypergraphs). For all f, r N wth f > r and all c, p (0, 1] wth ( ) q + r c 0.9(p/2) h /(q r 4 q ), where q := 2f f! and h := 2 r, r there exst n 0 N and γ > 0 such that the followng holds for all n n 0. Let F be any r-graph on f vertces and let λ N wth λ γn. Suppose that G s a (c, h, p)-typcal r-graph on n vertces. Then G has an (F, λ)-desgn f t s (F, λ)-dvsble. The man result n [15] s also stated n the settng of typcal r-graphs, but addtonally requres that c 1/h p, 1/f and that λ = O(1) and F s complete. The case when F s complete and λ s bounded s also a specal case of our recent result on desgns n supercomplexes (see Theorem 1.4 n [12]). Prevous results n the case when r 3 and F s not complete are very sporadc for nstance Hanan [13] settled the problem f F s an octahedron (vewed as a 3-unform hypergraph) and G s complete. As a very specal case, Theorem 1.1 resolves a conjecture of Archdeacon on self-dual embeddngs of random graphs n orentable surfaces: as proved n [2], a graph has such an embeddng f t has a decomposton nto K := K (2) 4 and K := K (2) 5. Suppose G s a (c, h, p)-typcal 2-graph on n vertces wth an even number of edges and 1/n c 1/h p (whch almost surely holds for the bnomal random graph G n,p f we remove at most one edge). Now remove a sutable number of copes of K from G to ensure that the leftover G satsfes 16 G. Let F be the vertex-dsjont unon of K and K. Snce Deg(F ) 1 = 1, G s F -dvsble. Thus we can apply

3 3 Theorem 1.1 to obtan an F -decomposton of G. If the number of edges s odd, a smlar argument yelds self-dual embeddngs n non-orentable surfaces. In Secton 8, we wll deduce Theorem 1.1 from a more general result on F -decompostons n supercomplexes G (Theorem 3.8). (The condton of G beng a supercomplex s consderably less restrctve than typcalty.) Moreover, the F -desgns we obtan wll have the addtonal property that V (F ) V (F ) r for all dstnct F, F whch are ncluded n the desgn. It s easy to see that wth ths addtonal property the bound on λ n Theorem 1.1 s best possble up to the value of γ. We can also deduce the followng result whch yelds near-optmal F -packngs n typcal r-graphs whch are not dvsble. (An F -packng n G s a collecton of edge-dsjont copes of F n G.) Theorem 1.2. For all f, r N wth f > r and all c, p (0, 1] wth c 0.9p h /(q r 4 q ), where q := 2f f! and h := 2 r ( q + r r there exst n 0, C N such that the followng holds for all n n 0. Let F be any r-graph on f vertces. Suppose that G s a (c, h, p)-typcal r-graph on n vertces. Then G has an F -packng F such that the leftover L consstng of all uncovered edges satsfes (L) C F -desgns n hypergraphs of large mnmum degree. Once the exstence queston s settled, a next natural step s to seek F -desgns and F -decompostons n r-graphs of large mnmum degree. Our next result gves a bound on the mnmum degree whch ensures an F -decomposton for weakly regular r-graphs F. These are defned as follows. Defnton 1.3 (weakly regular). Let F be an r-graph. We say that F s weakly (s 0,..., s r 1 )- regular f for all 0 r 1 and all S ( ) V (F ), we have F (S) {0, s }. We smply say that F s weakly regular f t s weakly (s 0,..., s r 1 )-regular for sutable s s. So for example, clques, the Fano plane and the octahedron are all weakly regular but a 3-unform tght or loose cycle s not. Theorem 1.4 (F -decompostons n hypergraphs of large mnmum degree). Let F be a weakly regular r-graph on f vertces. Let c r! F := 3 14 r f 2r. There exsts an n 0 N such that the followng holds for all n n 0. Suppose that G s an r-graph on n vertces wth δ(g) (1 c F )n. Then G has an F -decomposton f t s F -dvsble. Note that Theorem 1.4 mples that every packng of edge-dsjont copes of F nto K n (r) overall maxmum degree at most c F n can be extended nto an F -decomposton of K(r) n K (r) n ), wth (provded s F -dvsble). An analogous (but sgnfcantly worse) constant c F for r-graphs F whch are not weakly regular mmedately follows from the case p = 1 of Theorem 1.1. These results lead to the concept of the decomposton threshold δ F of a gven r-graph F. Defnton 1.5 (Decomposton threshold). Gven an r-graph F, let δ F be the nfmum of all δ [0, 1] wth the followng property: There exsts n 0 N such that for all n n 0, every F -dvsble r-graph G on n vertces wth δ(g) δn has an F -decomposton. By Theorem 1.4, we have δ F 1 c F whenever F s weakly regular. As noted n [12], for log f all r, f, n 0 N, there exsts an r-graph G n on n n 0 vertces wth δ(g n ) (1 b r )n such f r 1 that G n does not contan a sngle copy of K (r) f, where b r > 0 only depends on r. (Ths can be seen by adaptng a constructon from [19] whch s based on a result from [29].) Prevously, the only postve result for the hypergraph case r 3 was due to Yuster [36], who showed that f T s a lnear r-unform hypertree, then every T -dvsble r-graph G on n

4 4 S. GLOCK, D. KÜHN, A. LO AND D. OSTHUS vertces wth mnmum vertex degree at least ( 1 + o(1)) ( n 2 r 1) has a T -decomposton. Ths s r 1 asymptotcally best possble for nontrval T. Moreover, the result mples that δ T 1/2 r 1. For the graph case r = 2, much more s known about the decomposton threshold: the results n [4, 11] establsh a close connecton between δ F and the fractonal decomposton threshold δf (whch s defned as n Defnton 1.5, but wth an F -decomposton replaced by a fractonal F -decomposton). In partcular, the results n [4, 11] mply that δ F max{δf, 1 1/(χ(F )+1)} and that δ F = δf f F s a complete graph. Together wth recent results on the fractonal decomposton threshold for clques n [3, 8], ths gves the best current bounds on δ F for general F. It would be very nterestng to establsh a smlar connecton n the hypergraph case. Also, for bpartte graphs the decomposton threshold was completely determned n [11]. It would be nterestng to see f ths can be generalsed to r-partte r-graphs. On the other hand, even the decomposton threshold of a graph trangle s stll unknown (a beautful conjecture of Nash-Wllams [24] would mply that the value s 3/4) Countng. An approxmate F -decomposton of K n (r) s a set of edge-dsjont copes of F n K n (r) whch together cover almost all edges of K n (r). Gven good bounds on the number of approxmate F -decompostons of K n (r) whose set of leftover edges forms a typcal r-graph, one can apply Theorem 1.1 to obtan correspondng bounds on the number of F -decompostons n K n (r) (see [15, 16] for the clque case). Such bounds on the number of approxmate F - decompostons can be acheved by consderng ether a random greedy F -removal process or an assocated F -nbble removal process Outlne of the paper. As mentoned earler, our man result (Theorem 3.8) actually concerns F -decompostons n so-called supercomplexes. We wll defne supercomplexes n Secton 3 and derve Theorems 1.1, 1.2 and 1.4 from Theorem 3.8 n Secton 8. The defnton of a supercomplex G nvolves manly the dstrbuton of clques of sze f n G (where f = V (F ) ). The noton s weaker than usual notons of quasrandomness. Ths has two man advantages: frstly, our proof s by nducton on r, and workng wth ths weaker noton s essental to make the nducton proof work. Secondly, ths allows us to deduce Theorems 1.1, 1.2 and 1.4 from a sngle statement. However, Theorem 3.8 apples only to F -decompostons of a supercomplex G for weakly regular r-graphs F (whch allows us to deduce Theorem 1.4 but not Theorem 1.1). To deal wth ths, n Secton 8 we frst provde an explct constructon whch shows that every r-graph F can be perfectly packed nto a sutable weakly regular r-graph F. In partcular, F has an F -decomposton. The dea s then to apply Theorem 3.8 to fnd an F -decomposton n G. Unfortunately, G may not be F -dvsble. To overcome ths, n Secton 9 we show that we can remove a small set of copes of F from G to acheve that the leftover G of G s now F -dvsble (see Lemma 8.4 for the statement). Ths now mples Theorem 1.1 for F -decompostons,.e. for λ = 1. However, by repeatedly applyng Theorem 3.8 n a sutable way, we can actually allow λ to be as large as requred n Theorem 1.1. It thus remans to prove Theorem 3.8 tself. We acheve ths va the teratve absorpton method. The dea s to teratvely extend a packng of edge-dsjont copes of F untl the set H of uncovered edges s very small. Ths fnal set can then be absorbed nto an r-graph A we set asde at the begnnng of the proof (n the sense that A H has an F -decomposton). Ths teratve approach to decompostons was frst ntroduced n [18, 21] n the context of Hamlton decompostons of graphs. (Absorpton tself was poneered earler for spannng structures e.g. n [20, 28], but as remarked e.g. n [15], such drect approaches are not feasble n the decomposton settng.) Ths approach reles on beng able to fnd a sutable approxmate F -decomposton n each teraton, whose exstence we derve n Secton 5. The teraton process s underpnned by a so-called vortex, whch conssts of an approprate nested sequence of vertex subsets of G (after each teraton, the current set of uncovered edges s constraned to the next vertex subset n the

5 5 sequence). These vortces are dscussed n Secton 6. The fnal absorpton step s descrbed n Secton 7. As mentoned earler, the current proof bulds on the framework ntroduced n [12]. In fact, several parts of the argument n [12] can ether be used drectly or can be straghtforwardly adapted to the current settng. In partcular, ths apples to the Cover down lemma (Lemma 6.9), whch s the key result that allows the teraton to work. Thus n the current paper we concentrate on the parts whch nvolve sgnfcant new deas (e.g. the absorpton process). For detals of the parts whch can be straghtforwardly adapted, we refer to the appendx. Altogether, ths llustrates the versatlty of our framework and we thus beleve that t can be developed n further settngs. As a byproduct of the constructon of the weakly regular r-graph F outlned above, we prove the exstence of resolvable clque decompostons n complete partte r-graphs G (see Theorem 8.1). The constructon s explct and explots the property that all square submatrces of so-called Cauchy matrces over fnte felds are nvertble. We beleve ths constructon to be of ndependent nterest. A natural queston leadng on from the current work would be to obtan such resolvable decompostons also n the general (non-partte) case. For decompostons of K n (2) nto K (2), ths s due to Ray-Chaudhur and Wlson [26]. For recent progress see [9, 23]. f 2. Notaton 2.1. Basc termnology. We let [n] denote the set {1,..., n}, where [0] :=. Moreover, let [n] 0 := [n] {0} and N 0 := N {0}. As usual, ( n ) denotes the bnomal coeffcent, where we set ( ) n := 0 f > n or < 0. Moreover, gven a set X and N0, we wrte ( ) X for the collecton of all -subsets of X. Hence, ( ) X = f > X. If F s a collecton of sets, we defne F := f F f. We wrte A B for the unon of A and B f we want to emphasse that A and B are dsjont. We wrte X B(n, p) f X has bnomal dstrbuton wth parameters n, p, and we wrte bn(n, p, ) := ( n ) p (1 p) n. So by the above conventon, bn(n, p, ) = 0 f > n or < 0. We say that an event holds wth hgh probablty (whp) f the probablty that t holds tends to 1 as n (where n usually denotes the number of vertces). We wrte x y to mean that for any y (0, 1] there exsts an x 0 (0, 1) such that for all x x 0 the subsequent statement holds. Herarches wth more constants are defned n a smlar way and are to be read from the rght to the left. We wll always assume that the constants n our herarches are reals n (0, 1]. Moreover, f 1/x appears n a herarchy, ths mplctly means that x s a natural number. More precsely, 1/x y means that for any y (0, 1] there exsts an x 0 N such that for all x N wth x x 0 the subsequent statement holds. We wrte a = b ± c f b c a b + c. Equatons contanng ± are always to be nterpreted from left to rght, e.g. b 1 ± c 1 = b 2 ± c 2 means that b 1 c 1 b 2 c 2 and b 1 + c 1 b 2 + c 2. When dealng wth multsets, we treat multple appearances of the same element as dstnct elements. In partcular, two subsets A, B of a multset can be dsjont even f they both contan a copy of the same element, and f A and B are dsjont, then the multplcty of an element n the unon A B s obtaned by addng the multplctes of ths element n A and B (rather than just takng the maxmum) Hypergraphs and complexes. Let G be an r-graph. Note that G( ) = G. For a set S V (G) wth S r and L G(S), let S L := {S e : e L}. Clearly, there s a natural bjecton between L and S L. For [r 1] 0, we defne δ (G) and (G) as the mnmum and maxmum value of G(S) over all -subsets S of V (G), respectvely. As before, we let δ(g) := δ r 1 (G) and (G) := r 1 (G). Note that δ 0 (G) = 0 (G) = G( ) = G. For two r-graphs G and G, we let G G denote the r-graph obtaned from G by deletng all edges of G. We wrte G 1 + G 2 to mean the vertex-dsjont unon of G 1 and G 2, and t G to mean the vertex-dsjont unon of t copes of G.

6 6 S. GLOCK, D. KÜHN, A. LO AND D. OSTHUS Let F and G be r-graphs. An F -packng n G s a set F of edge-dsjont copes of F n G. We let F (r) denote the r-graph consstng of all covered edges of G,.e. F (r) = F F F. A mult-r-graph G conssts of a set of vertces V (G) and a multset of edges E(G), where each e E(G) s a subset of V (G) of sze r. We wll often dentfy a mult-r-graph wth ts edge set. For S V (G), let G(S) denote the number of edges of G that contan S (counted wth multplctes). If S = r, then G(S) s called the multplcty of S n G. We say that G s F -dvsble f Deg(F ) S dvdes G(S) for all S V (G) wth S r 1. An F -decomposton of G s a collecton F of copes of F n G such that every edge e G s covered precsely once. (Thus f S V (G) has sze r, then there are precsely G(S) copes of F n F n whch S forms an edge.) Defnton 2.1. A complex G s a hypergraph whch s closed under ncluson, that s, whenever e e G we have e G. If G s a complex and N 0, we wrte G () for the -graph on V (G) consstng of all e G wth e =. We say that a complex s empty f / G (0), that s, f G does not contan any edges. Suppose G s a complex and e V (G). Defne G(e) as the complex on vertex set V (G) \ e contanng all sets e V (G) \ e such that e e G. Clearly, f e / G, then G(e) s empty. Observe that f e = and r, then G (r) (e) = G(e) (r ). We say that G s a subcomplex of G f G s a complex and a subhypergraph of G. For a set U, defne G[U] as the complex on U V (G) contanng all e G wth e U. Moreover, for an r-graph H, let G[H] be the complex on V (G) wth edge set ( ) e G[H] := {e G : H}, r and defne G H := G[G (r) H]. So for [r 1], G[H] () = G (). For > r, we mght have G[H] () G (). Moreover, f H G (r), then G[H] (r) = H. Note that for an r 1 -graph H 1 and an r 2 -graph H 2, we have (G[H 1 ])[H 2 ] = (G[H 2 ])[H 1 ]. Also, (G H 1 ) H 2 = (G H 2 ) H 1, so we may wrte ths as G H 1 H 2. If G 1 and G 2 are complexes, we defne G 1 G 2 as the complex on vertex set V (G 1 ) V (G 2 ) contanng all sets e wth e G 1 and e G 2. We say that G 1 and G 2 are -dsjont f G () 1 G() 2 s empty. For any hypergraph H, let H be the complex on V (H) generated by H, that s, H := {e V (H) : e H such that e e }. For an r-graph H, we let H denote the complex on V (H) that s nduced by H, that s, ( ) e H := {e V (H) : H}. r Note that H (r) = H and for each [r 1] 0, H () s the complete -graph on V (H). We let K n denote the the complete complex on n vertces. 3. Decompostons of supercomplexes 3.1. Supercomplexes. We prove our man decomposton theorem for so-called supercomplexes, whch were ntroduced n [12]. The crucal property appearng n the defnton s that of regularty, whch means that every r-set of a gven complex G s contaned n roughly the same number of f-sets (where f = V (F ) ). If we vew G as a complex whch s nduced by some r-graph, ths means that every edge les n roughly the same number of clques of sze f. It turns out that ths set of condtons s approprate even when F s not a clque. A key advantage of the noton of a supercomplex s that the condtons are very flexble, whch wll enable us to boost ther parameters (see Lemma 3.5 below). The followng defntons are the same as n [12]. Defnton 3.1. Let G be a complex on n vertces, f N and r [f 1] 0, 0 ε, d, ξ 1. We say that G s

7 7 () (ε, d, f, r)-regular, f for all e G (r) we have G (f) (e) = (d ± ε)n f r ; () (ξ, f, r)-dense, f for all e G (r), we have G (f) (e) ξn f r ; () (ξ, f, r)-extendable, f G (r) s empty or there exsts a subset X V (G) wth X ξn such that for all e ( ) X r, there are at least ξn f r (f r)-sets Q V (G) \ e such that ) \ {e} G (r). ( Q e r We say that G s a full (ε, ξ, f, r)-complex f G s (ε, d, f, r)-regular for some d ξ, (ξ, f + r, r)-dense, (ξ, f, r)-extendable. We say that G s an (ε, ξ, f, r)-complex f there exsts an f-graph Y on V (G) such that G[Y ] s a full (ε, ξ, f, r)-complex. Note that G[Y ] (r) = G (r) (recall that r < f). Defnton 3.2. (supercomplex) Let G be a complex. We say that G s an (ε, ξ, f, r)-supercomplex f for every [r] 0 and every set B G () wth 1 B 2, we have that b B G(b) s an (ε, ξ, f, r )-complex. In partcular, takng = 0 and B = { } mples that every (ε, ξ, f, r)-supercomplex s also an (ε, ξ, f, r)-complex. Moreover, the above defnton ensures that f G s a supercomplex and b, b G (), then G(b) G(b ) s also a supercomplex (cf. Proposton 4.4). The followng examples from [12] demonstrate that the defnton of supercomplexes generalses the noton of typcalty. Example 3.3. Let 1/n 1/f and r [f 1]. Then the complete complex K n s a (0, 0.99/f!, f, r)- supercomplex. Example 3.4. Suppose that 1/n c, p, 1/f, that r [f 1] and that G s a (c, 2 r( ) f+r r, p)- typcal r-graph on n vertces. Then G s an (ε, ξ, f, r)-supercomplex, where ε := 2 f r+1 c/(f r)! and ξ := (1 2 f+1 c)p 2r ( f+r r ) /f!. As mentoned above, the followng lemma allows us to boost the regularty parameters (and thus deduce results wth effectve bounds). It s an easy consequence of our Boost lemma (Lemma 5.2). The key to the proof s that we can (probablstcally) choose some Y G (f) so that the parameters of G[Y ] n Defnton 3.1() are better than those of G,.e. the resultng dstrbuton of f-sets s more unform. Lemma 3.5 ([12]). Let 1/n ε, ξ, 1/f and r [f 1] wth 2(2 e) r ε ξ. Let ξ := 0.9(1/4) (f+r f ) ξ. If G s an (ε, ξ, f, r)-complex on n vertces, then G s an (n 1/3, ξ, f, r)-complex. In partcular, f G s an (ε, ξ, f, r)-supercomplex, then t s a (2n 1/3, ξ, f, r)-supercomplex The man complex decomposton theorem. The statement of our man complex decomposton theorem nvolves the concept of well separated decompostons. Ths dd not appear n [12], but s crucal for our nductve proof to work n the context of F -decompostons. Defnton 3.6 (well separated). Let F be an r-graph and let F be an F -packng (n some r-graph G). We say that F s κ-well separated f the followng hold: (WS1) for all dstnct F, F F, we have V (F ) V (F ) r. (WS2) for every r-set e, the number of F F wth e V (F ) s at most κ. We smply say that F s well separated f (WS1) holds.

8 8 S. GLOCK, D. KÜHN, A. LO AND D. OSTHUS For nstance, any K (r) f -packng s automatcally 1-well separated. Moreover, f an F -packng F s 1-well separated, then for all dstnct F, F F, we have V (F ) V (F ) < r. On the other hand, f F s not complete, we cannot requre V (F ) V (F ) < r n (WS1): ths would make t mpossble to fnd an F -decomposton of K n (r). The noton of beng well-separated s a natural relaxaton of ths requrement, we dscuss ths n more detal after statng Theorem 3.8. We now defne F -dvsblty and F -decompostons for complexes G (rather than r-graphs G). Defnton 3.7. Let F be an r-graph and f := V (F ). A complex G s F -dvsble f G (r) s F -dvsble. An F -packng n G s an F -packng F n G (r) such that V (F ) G (f) for all F F. Smlarly, we say that F s an F -decomposton of G f F s an F -packng n G and F (r) = G (r). Note that ths mples that every copy F of F used n an F -packng n G s supported by a clque,.e. G (r) [V (F )] = K (r) f. We can now state our man complex decomposton theorem. Theorem 3.8 (Man complex decomposton theorem). For all r N, the followng s true. ( ) r Let 1/n 1/κ, ε ξ, 1/f and f > r. Let F be a weakly regular r-graph on f vertces and let G be an F -dvsble (ε, ξ, f, r)-supercomplex on n vertces. Then G has a κ-well separated F -decomposton. We wll prove ( ) r by nducton on r n Secton 8. We do not make any attempt to optmse the values that we obtan for κ. We now motvate Defntons 3.6 and 3.7. Ths nvolves the followng addtonal concepts, whch are also convenent later. Defnton 3.9. Let f := V (F ) and suppose that F s a well separated F -packng. We let F denote the complex generated by the f-graph {V (F ) : F F}. We say that well separated F -packngs F 1, F 2 are -dsjont f F 1, F 2 are -dsjont (or equvalently, f V (F ) V (F ) < for all F F 1 and F F 2 ). Note that f F s a well-separated F -packng, then the f-graph {V (F ) : F F} s smple. Moreover, observe that (WS2) s equvalent to the condton r (F (f) ) κ. Furthermore, f F s a well separated F -packng n a complex G, then F s a subcomplex of G by Defnton 3.7. Clearly, we have F (r) F (r), but n general equalty does not hold. On the other hand, f F s an F -decomposton of G, then F (r) = G (r) whch mples F (r) = F (r). We now dscuss (WS1). Durng our proof, we wll need to fnd an F -packng whch covers a gven set of edges. Ths gves rse to the followng task of coverng down locally. ( ) Gven a set S V (G) of sze 1 r 1, fnd an F -packng F whch covers all edges of G that contan S. (Ths s crucal n the proof of Lemma 6.9. Moreover, a two-sded verson of ths nvolvng sets S, S s needed to construct parts of our absorbers, see Secton 7.1.) A natural approach to acheve ( ) s as follows: Let T ( ) V (F ). Suppose that by usng the man theorem nductvely, we can fnd an F (T )-decomposton F of G(S). We now wsh to obtan F by extendng F as follows: For each copy F of F (T ) n F, we defne a copy F of F by addng S back, that s, F has vertex set V (F ) S and S plays the role of T n F. Then F covers all edges e wth S e and e \ S F. Snce F s an F (T )-decomposton of G(S), the unon of all F would ndeed cover all edges of G that contan S, as desred. There are two ssues wth ths extenson though. Frstly, t s not clear that F s a subgraph of G. Secondly, for dstnct F, F F, t s not clear that F and F are edge-dsjont. Defnton 3.7 (and the succeedng remark) allows us to resolve the frst ssue. Indeed, f F s an F (T )-decomposton of the complex G(S), then from V (F ) G(S) (f ), we can deduce V (F ) G (f) and thus that F s a subgraph of G (r). We now consder the second ssue. Ths does not arse f F s a clque. Indeed, n that case F (T ) s a copy of K (r ) f, and thus for dstnct F, F F we have V (F ) V (F ) < r. Hence V (F ) V (F ) < r + S = r,.e. F and F are edge-dsjont. If however F s not a clque, then F, F F can overlap n r or more vertces (they could n fact have the same

9 9 vertex set), and the above argument does not work. We wll show that under the assumpton that F s well separated, we can overcome ths ssue and stll carry out the above extenson. (Moreover, the resultng F -packng F wll n fact be well separated tself, see Defnton 6.10 and Proposton 6.11). For ths t s useful to note that F (T ) s an (r )-graph, and thus we already have V (F ) V (F ) r f F s well separated. The reason why we also nclude (WS2) n Defnton 3.6 s as follows. Suppose we have already found a well separated F -packng F 1 n G and now want to fnd another well separated F -packng F 2 such that we can combne F 1 and F 2. If we fnd F 2 n G F (r) (r) 1, then F 1 and F (r) 2 are edge-dsjont and thus F 1 F 2 wll be an F -packng n G, but t s not necessarly well separated. We therefore fnd F 2 n G F (r) 1 F (r+1) 1. Ths ensures that F 1 and F 2 are (r + 1)- dsjont, whch n turn mples that F 1 F 2 s ndeed well separated, as requred. But n order to be able to construct F 2, we need to ensure that G F (r) 1 F (r+1) 1 s stll a supercomplex, whch s true f (F (r) (r+1) 1 ) and (F1 ) are small (cf. Proposton 4.5). The latter n turn s ensured by (WS2) va Fact 4.3. Fnally, we dscuss why we prove Theorem 3.8 for weakly regular r-graphs F. Most mportantly, the regularty of the degrees wll be crucal for the constructon of our absorbers (most notably n Lemma 7.22). Beyond that, weakly regular graphs also have useful closure propertes (cf. Proposton 4.2): they are closed under takng lnk graphs and dvsblty s nherted by lnk graphs n a natural way. We prove Theorem 3.8 n Sectons 5 7 and 8.1. As descrbed n Secton 1.5, we generalse ths to arbtrary F va Lemma 8.2 (proved n Secton 8.2) and Lemma 8.4 (proved n Secton 9): Lemma 8.2 shows that for every gven r-graph F, there s a weakly regular r-graph F whch has an F -decomposton. Lemma 8.4 then complements ths by showng that every F -dvsble r-graph G can be transformed nto an F -dvsble r-graph G by removng a sparse F -decomposable subgraph of G. 4. Tools 4.1. Basc tools. We wll often use the followng handshakng lemma for r-graphs: Let G be an r-graph and 0 k r 1. Then for every S ( ) V (G) we have ( ) r 1 (4.1) G(S) = G(T ). r k T ( V (G) k ): S T Proposton 4.1. Let F be an r-graph. Then there exst nfntely many n N such that K (r) n s F -dvsble. Proof. Let p := r 1 =0 Deg(F ). We wll show that for every a N, f we let n = r!ap + r 1 then K n (r) s F -dvsble. Clearly, ths mples the clam. In order to see that K n (r) s F -dvsble, t s suffcent to show that p ( n r ) for all [r 1]0. It s easy to see that ths holds for the above choce of n. The followng proposton shows that the class of weakly regular unform hypergraphs s closed under takng lnk graphs. Proposton 4.2. Let F be a weakly regular r-graph and let [r 1]. Suppose that S ( ) V (F ) and that F (S) s non-empty. Then F (S) s a weakly regular (r )-graph and Deg(F (S)) j = Deg(F ) +j for all j [r 1] 0. Proof. Let s 0,..., s r 1 be such that F s weakly (s 0,..., s r 1 )-regular. Note that snce F s nonempty, we have s j > 0 for all j [r 1] 0 (and the s s are unque). Consder j [r 1] 0. For all T ( ) V (F (S)) j, we have F (S)(T ) = F (S T ) {0, s+j }. Hence, F (S) s weakly (s,..., s r 1 )- regular. Snce F s non-empty, we have Deg(F ) = (s 0,..., s r 1 ), and snce F (S) s non-empty too by assumpton, we have Deg(F (S)) = (s,..., s r 1 ). Therefore, Deg(F (S)) j = Deg(F ) +j for all j [r 1] 0.

10 10 S. GLOCK, D. KÜHN, A. LO AND D. OSTHUS We now lst some useful propertes of well separated F -packngs. Fact 4.3. Let G be a complex and F an r-graph on f > r vertces. Suppose that F s a κ-well separated F -packng (n G) and F s a κ -well separated F -packng (n G). Then the followng hold. () (F (r+1) ) κ(f r). () If F (r) and F (r) are edge-dsjont and F and F are (r + 1)-dsjont, then F F s a (κ + κ )-well separated F -packng (n G). () If F and F are r-dsjont, then F F s a max{κ, κ }-well separated F -packng (n G) Some propertes of supercomplexes. The followng propertes of supercomplexes were proved n [12]. Proposton 4.4 ([12]). Let G be an (ε, ξ, f, r)-supercomplex and let B G () wth 1 B 2 for some [r] 0. Then b B G(b) s an (ε, ξ, f, r )-supercomplex. Proposton 4.5 ([12]). Let f, r N and r N 0 wth f > r and r r. Let G be a complex on n r2 r+1 vertces and let H be an r -graph on V (G) wth (H) γn. Then the followng hold: () If G s (ε, d, f, r)-regular, then G H s (ε + 2 r γ, d, f, r)-regular. () If G s (ξ, f, r)-dense, then G H s (ξ 2 r γ, f, r)-dense. () If G s (ξ, f, r)-extendable, then G H s (ξ 2 r γ, f, r)-extendable. (v) If G s an (ε, ξ, f, r)-complex, then G H s an (ε + 2 r γ, ξ 2 r γ, f, r)-complex. (v) If G s an (ε, ξ, f, r)-supercomplex, then G H s an (ε + 2 2r+1 γ, ξ 2 2r+1 γ, f, r)- supercomplex. Corollary 4.6 ([12]). Let 1/n ε, γ, ξ, p, 1/f and r [f 1]. Let ( f+r ) ξ := 0.95ξp 2r ( f+r r ) 0.95ξp (8 f ) and γ := r r (f r)! γ. Suppose that G s an (ε, ξ, f, r)-supercomplex on n vertces and that H G (r) s a random subgraph obtaned by ncludng every edge of G (r) ndependently wth probablty p. Then whp the followng holds: for all L G (r) wth (L) γn, G[H L] s a (3ε + γ, ξ γ, f, r)- supercomplex Rooted Embeddngs. We now prove a result (Lemma 4.7) whch allows us to fnd edgedsjont embeddngs of graphs wth a prescrbed root embeddng. Let T be an r-graph and suppose that X V (T ) s such that T [X] s empty. A root of (T, X) s a set S X wth S [r 1] and T (S) > 0. For an r-graph G, we say that Λ: X V (G) s a G-labellng of (T, X) f Λ s njectve. Our am s to embed T nto G such that the roots of (T, X) are embedded at ther assgned poston. More precsely, gven a G-labellng Λ of (T, X), we say that φ s a Λ-fathful embeddng of (T, X) nto G f φ s an njectve homomorphsm from T to G wth φ X = Λ. Moreover, for a set S V (G) wth S [r 1], we say that Λ roots S f S Im(Λ) and T (Λ 1 (S)) > 0,.e. f Λ 1 (S) s a root of (T, X). The degeneracy of T rooted at X s the smallest D such that there exsts an orderng v 1,..., v k of the vertces of V (T ) \ X such that for every l [k], we have T [X {v 1,..., v l }](v l ) D,.e. every vertex s contaned n at most D edges whch le to the left of that vertex n the orderng. We need to be able to embed many copes of (T, X) smultaneously (wth dfferent labellngs) nto a gven host graph G such that the dfferent embeddngs are edge-dsjont. In fact, we need a slghtly stronger dsjontness crteron. Ideally, we would lke to have that two dstnct embeddngs ntersect n less than r vertces. However, ths s n general not possble because of the desred rootng. We therefore ntroduce the followng concept of a hull. We wll ensure that the hulls are edge-dsjont, whch wll be suffcent for our purposes. Gven (T, X) as above, the

11 11 hull of (T, X) s the r-graph T on V (T ) wth e T f and only f e X = or e X s a root of (T, X). Note that T T K (r) V (T ) K(r) X, where K(r) Z denotes the complete r-graph wth vertex set Z. Moreover, the roots of (T, X) are precsely the roots of (T, X). Lemma 4.7. Let 1/n γ ξ, 1/t, 1/D and r [t]. Suppose that α (0, 1] s an arbtrary scalar (whch mght depend on n) and let m αγn r be an nteger. For every j [m], let T j be an r-graph on at most t vertces and X j V (T j ) such that T j [X j ] s empty and T j has degeneracy at most D rooted at X j. Let G be an r-graph on n vertces such that for all A ( ) V (G) r 1 wth A D, we have S A G(S) ξn. Let O be an (r + 1)-graph on V (G) wth (O) γn. For every j [m], let Λ j be a G-labellng of (T j, X j ). Suppose that for all S V (G) wth S [r 1], we have that (4.2) {j [m] : Λ j roots S} αγn r S 1. Then for every j [m], there exsts a Λ j -fathful embeddng φ j of (T j, X j ) nto G such that the followng hold: () for all dstnct j, j [m], the hulls of (φ j (T j ), Im(Λ j )) and (φ j (T j ), Im(Λ j )) are edgedsjont; () for all j [m] and e O wth e Im(φ j ), we have e Im(Λ j ); () ( j [m] φ j(t j )) αγ (2 r) n. Note that () mples that φ 1 (T 1 ),..., φ m (T m ) are edge-dsjont. We also remark that the T j do not have to be dstnct; n fact, they could all be copes of a sngle r-graph T. Proof. For j [m] and a set S V (G) wth S [r 1], let root(s, j) := {j [j] : Λ j roots S}. We wll defne φ 1,..., φ m successvely. Once φ j s defned, we let K j denote the hull of (φ j (T j ), Im(Λ j )). Note that φ j (T j ) K j and that K j s not necessarly a subgraph of G. Suppose that for some j [m], we have already defned φ 1,..., φ j 1 such that K 1,..., K j 1 are edge-dsjont, () holds for all j [j 1], and the followng holds for G j := j [j 1] K j, all [r 1] and all S ( V (G) (4.3) ) : G j (S) αγ (2 ) n r + (root(s, j 1) + 1)2 t. Note that (4.3) together wth (4.2) mples that for all [r 1] and all S ( V (G) (4.4) G j (S) 2αγ (2 ) n r. ), we have We wll now defne a Λ j -fathful embeddng φ j of (T j, X j ) nto G such that K j s edge-dsjont from G j, () holds for j, and (4.3) holds wth j replaced by j + 1. For [r 1], defne BAD := {S ( ) V (G) : Gj (S) αγ (2 ) n r }. We vew BAD as an -graph. We clam that for all [r 1], (4.5) (BAD ) γ (2 r) n. Consder [r 1] and suppose that there exsts some S ( ) V (G) 1 such that BAD (S) > γ (2 r) n. We then have that 1 G j (S) = G j (S {v}) r 1 G j (S {v}) r + 1 v V (G)\S v BAD (S) r 1 BAD (S) αγ (2 ) n r r 1 γ (2 r) nαγ (2 ) n r = r 1 αγ (2 r +2 ) n r ( 1). Ths contradcts (4.4) f 1 > 0 snce 2 r + 2 < 2 ( 1). If = 1, then S = and we have G j r 1 αγ (2 r +2 1) n r, whch s also a contradcton snce G j m ( ( t r) t r) αγn r and 2 r < 1 (as r 2 f [r 1]). Ths proves (4.5). We now embed the vertces of T j such that the obtaned embeddng φ j s Λ j -fathful. Frst, embed every vertex from X j at ts assgned poston. Snce T j has degeneracy at most D rooted

12 12 S. GLOCK, D. KÜHN, A. LO AND D. OSTHUS at X j, there exsts an orderng v 1,..., v k of the vertces of V (T j ) \ X j such that for every l [k], we have (4.6) T j [X j {v 1,..., v l }](v l ) D. Suppose that for some l [k], we have already embedded v 1,..., v l 1. We now want to defne φ j (v l ). Let U := {φ j (v) : v X j {v 1,..., v l 1 }} be the set of vertces whch have already been used as mages for φ j. Let A contan all (r 1)-subsets S of U such that φ 1 j (S) {v l } T j. We need to choose φ j (v l ) from the set ( S A G(S)) \ U n order to complete φ j to an njectve homomorphsm from T j to G. By (4.6), we have A D. Thus, by assumpton, S A G(S) ξn. For [r 1], let O consst of all vertces x V (G) such that there exsts some S ( ) U 1 such that S {x} BAD (so BAD 1 = ( O 1 ) 1 ). We have O ( ) U (BAD ) (4.5) 1 ( t 1 ) γ (2 r) n. Let O r consst of all vertces x V (G) such that S {x} G j for some S ( U r 1). By (4.4), we have that O r ( ) U r 1 (Gj ) ( ) t r 1 2αγ (2 (r 1)) n ( t r 1) γ (2 r) n. Fnally, let O r+1 be the set of all vertces x V (G) such that there exsts some S ( ) U r such that S {x} O. By assumpton, we have O r+1 ( ) ( U r (O) t ) r γn. Crucally, we have S A r+1 G(S) U O ξn t 2 t γ (2 r) n > 0. =1 Thus, there exsts a vertex x V (G) such that x / U O 1 O r+1 and S {x} G for all S A. Defne φ j (v l ) := x. Contnung n ths way untl φ j s defned for every v V (T j ) yelds an njectve homomorphsm from T j to G. By defnton of O r+1, () holds for j. Moreover, by defnton of O r, K j s edge-dsjont from G j. It remans to show that (4.3) holds wth j replaced by j +1. Let [r 1] and S ( ) V (G). If S / BAD, then we have G j+1 (S) G j (S) + ( t r ) αγ (2 ) n r + 2 t, so (4.3) holds. Now, assume that S BAD. If S Im(Λ j ) and T j (Λ 1 j (S)) > 0, then root(s, j) = root(s, j 1) + 1 and thus G j+1 (S) G j (S) + ( t r ) αγ (2 ) n r + (root(s, j 1)+1)2 t + ( t r ) αγ (2 ) n r +(root(s, j)+1)2 t and (4.3) holds. Suppose next that S Im(Λ j ). We clam that S V (φ j (T j )). Suppose, for a contradcton, that S V (φ j (T j )). Let l := max{l [k] : φ j (v l ) S}. (Note that the maxmum exsts snce (S V (φ j (T j ))) \ Im(Λ j ) s not empty.) Hence, x := φ j (v l ) S. Recall that when we defned φ j (v l ), φ j (v) had already been defned for all v X j {v 1,..., v l 1 } and hence S \ {x} U. But snce S BAD, we have x O, n contradcton to x = φ j (v l ). Thus, S V (φ j (T j )) = V (K j ), whch clearly mples that G j+1 (S) = G j (S) and (4.3) holds. The last remanng case s f S Im(Λ j ) but T j (Λ 1 j (S)) = 0. But then S s not a root of (φ j (T j ), Im(Λ j )) and thus not a root of (K j, Im(Λ j )). Hence K j (S) = 0 and therefore G j+1 (S) = G j (S) as well. Fnally, f j = m, then the fact that (4.3) holds wth j replaced by j + 1 together wth (4.2) mples that ( j [m] φ j(t j )) 2αγ (2 (r 1)) n αγ (2 r) n. 5. Approxmate F -decompostons The majorty of the edges whch are covered durng our teratve absorpton procedure are covered by approxmate F -decompostons of certan parts of G,.e. F -packngs whch cover almost all the edges n these parts. For clques, the exstence of such packngs was frst proved by Rödl [27], ntroducng what s now called the nbble technque. Here, we derve a result on approxmate F -decompostons whch s sutable for our needs (Lemma 5.3).

13 13 We wll derve the F -nbble lemma (Lemma 5.3) from the specal case when F s a clque. Ths n turn was derved n [12] from a result n [1] whch allows us to assume that the leftover of an approxmate clque decomposton has approprately bounded maxmum degree. Lemma 5.1 (Boosted nbble lemma, [12]). Let 1/n γ, ε ξ, 1/f and r [f 1]. Let G be a complex on n vertces such that G s (ε, d, f, r)-regular and (ξ, f + r, r)-dense for some d ξ. Then G contans a K (r) f -packng K such that (G(r) K (r) ) γn. Crucally, we do not need to assume that ε γ n Lemma 5.1. The reason for ths s the socalled Boost lemma from [12], whch allows us to boost the regularty parameters of a sutable complex and whch s an mportant ngredent n the proof of both Lemma 5.1 and Lemma 5.3. Lemma 5.2 (Boost lemma, [12]). Let 1/n ε, ξ, 1/f and r [f 1] such that 2(2 e) r ε ξ. Let ξ := 0.9(1/4) (f+r f ) ξ. Suppose that G s a complex on n vertces and that G s (ε, d, f, r)- regular and (ξ, f + r, r)-dense for some d ξ. Then there exsts Y G (f) such that G[Y ] s (n (f r)/2.01, d/2, f, r)-regular and (ξ, f + r, r)-dense. We now prove an F -nbble lemma whch allows us to fnd κ-well separated approxmate F - decompostons n supercomplexes. Lemma 5.3 (F -nbble lemma). Let 1/n 1/κ γ, ε ξ, 1/f and r [f 1]. Let F be an r-graph on f vertces. Let G be a complex on n vertces such that G s (ε, d, f, r)-regular and (ξ, f + r, r)-dense for some d ξ. Then G contans a κ-well separated F -packng F such that (G (r) F (r) ) γn. Let F be an r-graph on f vertces. Gven a collecton K of edge-dsjont copes of K (r) f, we defne the K-random F -packng F as follows: For every K K, choose a random bjecton from V (F ) to V (K) and let F K be a copy of F on V (K) embedded by ths bjecton. Let F := {F K : K K}. Clearly, f K s a K (r) f -decomposton of a complex G, then the K-random F -packng F s a 1-well separated F -packng n G. Moreover, wrtng p := 1 F / ( f r), we have F (r) = F K = F G (r) / ( f r) = (1 p) G (r), and for every e G (r), we have P(e G (r) F (r) ) = p. As turns out, the leftover G (r) F (r) behaves essentally lke a p-random subgraph of G (r) (cf. Lemma 5.4). Our strategy to prove Lemma 5.3 s thus as follows: We apply Lemma 5.1 to G to obtan a -packng K 1 such that (G (r) K (r) ) γn. The leftover here s neglgble, so assume for K (r) f 1 the moment that K 1 s a K (r) f -decomposton. We then choose a K 1-random F -packng F 1 n G and contnue the process wth G F (r) 1. In each step, the leftover decreases by a factor of p. Thus after log p γ steps, the leftover wll have maxmum degree at most γn. Lemma 5.4. Let 1/n ε ξ, 1/f and r [f 1]. Let F be an r-graph on f-vertces wth p := 1 F / ( f r) (0, 1). Let G be an (ε, d, f, r)-regular and (ξ, f + r, r)-dense complex on n vertces for some d ξ. Suppose that K s a K (r) f -decomposton of G. Let F be the K-random F -packng n G. Then whp the followng hold for G := G K (r+1) F (r). () G s (2ε, p (f r) 1 d, f, r)-regular; () G s (0.9p (f+r r ) 1 ξ, f + r, r)-dense; () (G (r) ) 1.1p (G (r) ). Snce the assertons follow easly from the defntons, we omt the proof here. We refer to Appendx A for the detals. Proof of Lemma 5.3. Let p := 1 F / ( ) f (r) r. If F = K f, then we are done by Lemma 5.1. We may thus assume that p (0, 1). Choose ε > 0 such that 1/n ε 1/κ γ, ε p, 1 p, ξ, 1/f. We wll now repeatedly apply Lemma 5.1. More precsely, let ξ 0 := 0.9(1/4) (f+r f ) ξ and defne ξ j := (0.5p) j(f+r r ) ξ0 for j 1. For every j [κ] 0, we wll fnd F j and G j such that the followng hold:

14 14 S. GLOCK, D. KÜHN, A. LO AND D. OSTHUS (a) j F j s a j-well separated F -packng n G and G j G F (r) (b) j (L j ) jε n, where L j := G (r) F (r) j G (r) j ; (c) j G j s (2 (r+1)j ε, d j, f, r)-regular and (ξ j, f + r, r)-dense for some d j ξ j ; (d) j F j and G j are (r + 1)-dsjont; (e) j (G (r) j ) (1.1p) j n. Frst, apply Lemma 5.2 to G n order to fnd Y G (f) such that G 0 := G[Y ] s (ε, d/2, f, r)- regular and (ξ 0, f + r, r)-dense. Hence, (a) 0 (e) 0 hold wth F 0 :=. Also note that F κ wll be a κ-well separated F -packng n G and (G (r) F (r) κ j ; ) (L κ )+ (G (r) κ ) κε n+(1.1p) κ n γn, so we can take F := F κ. Now, assume that for some j [κ], we have found F j 1 and G j 1 and now need to fnd F j and G j. By (c) j 1, G j 1 s ( ε, d j 1, f, r)-regular and (ξ j 1, f + r, r)-dense for some d j 1 ξ j 1. Thus, we can apply Lemma 5.1 to obtan a K (r) f -packng K j n G j 1 such that (L j ) ε n, where L j := G(r) j 1 K(r) j. Let G j := G j 1 L j. Clearly, K j s a K (r) f -decomposton of G j. Moreover, by (c) j 1 and Proposton 4.5 we have that G j s (2(r+1)(j 1)+r ε, d j 1, f, r)-regular and (0.9ξ j 1, f + r, r)-dense. By Lemma 5.4, there exsts a 1-well separated F -packng F j n G j such that the followng hold for G j := G j F (r) j K (r+1) j () G j s (2 (r+1)(j 1)+r+1 ε, p (f r) 1 d j 1, f, r)-regular; () G j s (0.81p (f+r r ) 1 ξ j 1, f + r, r)-dense; () (G (r) j ) 1.1p (G (r) j ). Let F j := F j 1 F j and L j := G (r) F (r) j = G j F (r) j F (r+1) j : G (r) j. Note that F (r) j 1 F (r) j = by (a) j 1. Moreover, F j 1 and F j are (r + 1)-dsjont by (d) j 1. Thus, F j s (j 1 + 1)-well separated by Fact 4.3(). Moreover, usng (a) j 1, we have G j G j 1 F (r) j G F (r) j 1 F (r) j, thus (a) j holds. Observe that L j \L j 1 L j. Thus, we clearly have (L j) (L j 1 )+ (L j ) jε n, so (b) j holds. Moreover, (c) j follows drectly from () and (), and (e) j follows from (e) j 1 and (). To see (d) j, observe that F j 1 and G j are (r + 1)-dsjont by (d) j 1 and snce G j G j 1, and F j and G j are (r + 1)-dsjont by defnton of G j. Thus, (a) j (e) j hold and the proof s completed. 6. Vortces A vortex s best thought of as a sequence of nested random-lke subsets of the vertex set of a supercomplex G. In our approach, the fnal set of the vortex has bounded sze. The man results of ths secton are Lemmas 6.3 and 6.4, where the frst one shows that vortces exst, and the latter one shows that gven a vortex, we can fnd an F -packng coverng all edges whch do not le nsde the fnal vortex set. We now gve the formal defnton of what t means to be a random-lke subset. Defnton 6.1. Let G be a complex on n vertces. We say that U s (ε, µ, ξ, f, r)-random n G f there exsts an f-graph Y on V (G) such that the followng hold: (R1) U V (G) wth U = µn ± n 2/3 ; (R2) there exsts d ξ such that for all x [f r] 0 and all e G (r), we have that {Q G[Y ] (f) (e) : Q U = x} = (1 ± ε)bn(f r, µ, x)dn f r ; (R3) for all e G (r) we have G[Y ] (f+r) (e)[u] ξ(µn) f ; (R4) for all h [r] 0 and all B G (h) wth 1 B 2 h we have that b B G(b)[U] s an (ε, ξ, f h, r h)-complex.

15 15 Havng defned what t means to be a random-lke subset, we can now defne what a vortex s. Defnton 6.2 (Vortex). Let G be a complex. An (ε, µ, ξ, f, r, m)-vortex n G s a sequence U 0 U 1 U l such that (V1) U 0 = V (G); (V2) U = µ U 1 for all [l]; (V3) U l = m; (V4) for all [l], U s (ε, µ, ξ, f, r)-random n G[U 1 ]; (V5) for all [l 1], U \ U +1 s (ε, µ(1 µ), ξ, f, r)-random n G[U 1 ]. As shown n [12], a vortex can be found n a supercomplex by repeatedly takng random subsets. Lemma 6.3 ([12]). Let 1/m ε µ, ξ, 1/f such that µ 1/2 and r [f 1]. Let G be an (ε, ξ, f, r)-supercomplex on n m vertces. Then there exsts a (2 ε, µ, ξ ε, f, r, m)-vortex n G for some µm m m. The followng s the man lemma of ths secton. Gven a vortex n a supercomplex G, t allows us to cover all edges of G (r) except possbly some from nsde the fnal vortex set (see Lemma 7.13 n [12] for the correspondng result n the case when F s a clque). Lemma 6.4. Let 1/m 1/κ ε µ ξ, 1/f and r [f 1]. Assume that ( ) k s true for all k [r 1]. Let F be a weakly regular r-graph on f vertces. Let G be an F -dvsble (ε, ξ, f, r)-supercomplex and U 0 U 1 U l an (ε, µ, ξ, f, r, m)-vortex n G. Then there exsts a 4κ-well separated F -packng F n G whch covers all edges of G (r) except possbly some nsde U l. The proof of Lemma 6.4 conssts of an teratve absorpton procedure, where the key ngredent s the Cover down lemma (Lemma 6.9). Roughly speakng, gven a supercomplex G and a random-lke subset U V (G), the Cover down lemma allows us to fnd a partal absorber H G (r) such that for any sparse L G (r), H L has an F -packng whch covers all edges of H L except possbly some nsde U. Together wth the F -nbble lemma (Lemma 5.3), ths allows us to cover all edges of G except possbly some nsde U whlst usng only few edges nsde U. Indeed, set asde H as above, whch s reasonably sparse. Then apply the Lemma 5.3 to G G (r) [U] H to obtan an F -packng F nbble wth a very sparse leftover L. Combne H and L to fnd an F -packng F clean whose leftover les nsde U. Now, f U 0 U 1 U l s a vortex, then U 1 s random-lke n G and thus we can cover all edges whch are not nsde U 1 by usng only few edges nsde U 1 (and n ths step we forbd edges nsde U 2 from beng used.) Then U 2 s stll random-lke n the remander of G[U 1 ], and hence we can terate untl we have covered all edges of G except possbly some nsde U l. The proof of Lemma 6.4 s very smlar to that of Lemma 7.13 n [12], thus we omt t here. The detals can be found n Appendx B. We record the followng easy tools from [12] for later use. Fact 6.5 ([12]). The followng hold. () If G s an (ε, ξ, f, r)-supercomplex, then V (G) s (ε/ξ, 1, ξ, f, r)-random n G. () If U s (ε, µ, ξ, f, r)-random n G, then G[U] s an (ε, ξ, f, r)-supercomplex. Proposton 6.6 ([12]). Let 1/n ε µ 1, µ 2, 1 µ 2, ξ, 1/f and r [f 1]. Let G be a complex on n vertces and let U V (G) be of sze µ 1 n and (ε, µ 1, ξ, f, r)-random n G. Then there exsts Ũ U of sze µ 2 U such that () Ũ s (ε + U 1/6, µ 2, ξ U 1/6, f, r)-random n G[U] and () U \ Ũ s (ε + U 1/6, µ 1 (1 µ 2 ), ξ U 1/6, f, r)-random n G. Proposton 6.7 ([12]). Let 1/n ε µ, ξ, 1/f such that µ 1/2 and r [f 1]. Suppose that G s a complex on n vertces and U s (ε, µ, ξ, f, r)-random n G. Suppose that L G (r) and O G (r+1) satsfy (L) εn and (O) εn. Then U s stll ( ε, µ, ξ ε, f, r)-random n G L O.