Spanning structures and universality in sparse hypergraphs
aa r X i v : . [ m a t h . C O ] A p r SPANNING STRUCTURES AND UNIVERSALITY IN SPARSEHYPERGRAPHS
OLAF PARCZYK AND YURY PERSON
Abstract.
In this paper the problem of finding various spanning structures in random hyper-graphs is studied. We notice that a general result of Riordan [
Spanning subgraphs of randomgraphs , Combinatorics, Probability & Computing (2000), no. 2, 125–148] can be adaptedfrom random graphs to random r -uniform hypergaphs and provide sufficient conditions whena random r -uniform hypergraph H ( r ) ( n, p ) contains a given spanning structure a.a.s. We alsodiscuss several spanning structures such as cube-hypergraphs, lattices, spheres and Hamiltoncycles in hypergraphs.Moreover, we study universality, i.e. when does an r -uniform hypergraph contain any hyper-graph on n vertices and with maximum vertex degree bounded by ∆? For H ( r ) ( n, p ) it is shownthat this holds for p = ω (cid:16) (ln n/n ) / ∆ (cid:17) a.a.s. by combining approaches taken by Dellamonica,Kohayakawa, R¨odl and Ruci´nski [ An improved upper bound on the density of universal randomgraphs , Random Structures Algorithms (2015), no. 2, 274–299] and of Ferber, Nenadov andPeter [ Universality of random graphs and rainbow embedding , Random Structures Algorithms,to appear]. Furthermore it is shown that the random graph G ( n, p ) for appropriate p andexplicit constructions due to Alon, Capalbo, Kohayakawa, R¨odl, Ruci´nski and Szemer´edi [7]and Alon and Capalbo [4, 5] of universal graphs yield constructions of universal hypergraphsthat are sparser than the random hypergraph H ( r ) ( n, p ) with p = ω (cid:16) (ln n/n ) / ∆ (cid:17) . Introduction
Finding spanning subgraphs is a well studied problem in random graph theory, see e.g. thefollowing monographs on random graphs [9, 23]. In the case of hypergraphs less is known andit is natural to study the corresponding spanning problems for random hypergraphs.An r -uniform hypergraph H is a tuple ( V, E ), where V is its vertex set and E ⊆ (cid:0) Vr (cid:1) theset of edges in H . The random r -uniform hypergraph H ( r ) ( n, p ) is the probability space of alllabelled r -uniform hypergraphs with the vertex set [ n ] := { , , . . . , n } where each edge e ∈ (cid:0) [ n ] r (cid:1) is chosen independently of all the other edges with probability p . Thus, for r = 2 this is astandard model of the random graph G ( n, p ). Let H = H ( i ) be a sequence of fixed r -uniformhypergraphs with n vertices, where n = n ( i ) → ∞ . Then we say that H ( r ) ( n, p ) contains thehypergraph H asymptotically almost surely (a.a.s.) if the probability that H ( i ) ⊆ H ( r ) ( n, p )tends to 1 as n tends to infinity. We say that ˆ p is a threshold function if P [ H ⊆ H ( r ) ( n, p )] → p ≪ ˆ p and P [ H ⊆ H ( r ) ( n, p )] → p ≫ ˆ p as n tends to infinity.It was shown by Bollob´as and Thomason [10] that all nontrivial monotone properties have athreshold function. Since subgraph containment is a monotone property it is natural to studythe threshold functions for appearance of various structures in random graphs and hypergraphs.The purpose of this paper is to obtain generalizations of several results about spanning struc-tures in random graphs to random hypergraphs. In particular we extend a result of Riordan [37]on containment of a given single spanning graph in G ( n, p ) and we generalize a result of Del-lamonica, Kohayakawa, R¨odl and Ruci´nski [13] on universality for the class of bounded degreespanning subgraphs in G ( n, p ) to random hypergraph H ( r ) ( n, p ). Furthermore, we show thatthe random graph G ( n, p ) for appropriate p and explicit constructions of universal graphs due toAlon, Capalbo, Kohayakawa, R¨odl, Ruci´nski and Szemer´edi [7] and Alon and Capalbo [4, 5] yield Date : September 5, 2018.This research was supported by DFG grant PE 2299/1-1. onstructions of universal hypergraphs that are sparser than the random hypergraph H ( r ) ( n, p )with p = ω (cid:0) (ln n/n ) / ∆ (cid:1) .1.1. Single spanning structures.
First spanning structures considered in graphs were perfectmatchings [18] and Hamilton cycles [30, 36] (see also [9, Chapter 8] and references therein). Morerecently, the thresholds for the appearance of (bounded degree) spanning trees [3, 31, 22, 20,24, 26, 27] were studied as well, for the currently best bounds see [33, 34].Alon and F¨uredi [8] studied the question when the random graph G ( n, p ) contains a givengraph G of bounded maximum degree ∆ hereby proving the bound p ≥ C (ln n/n ) / ∆ for someabsolute constant C >
0. In [37] Riordan proved quite a general theorem applicable to var-ious graphs in particular determining the threshold functions for the appearance of spanninghypercubes and lattices. K¨uhn and Osthus [32] determined an approximate threshold for theappearance of a square of a Hamilton cycle. Finding thresholds for factors of graphs and hyper-graphs was long an open problem where breakthrough was achieved by Johansson, Kahn andVu [25]. Kahn and Kalai [28] have a general conjecture about the thresholds for the appearanceof a given structure (which roughly states that the threshold p with P ( G ⊆ G ( n, p )) = 1 / G is within a factor of O (ln n ) from p E at which the expected number of copiesof G in G ( n, p E ) is 1, where p E is the so-called expectation threshold).When one turns to hypergraphs, so apart from perfect matchings and general factors [25],the only other spanning structures that were studied more recently are Hamilton cycles . An ℓ -overlapping Hamilton cycle is an r -uniform hypergraph such that for some cyclic ordering of[ n ] and an ordering of the edges, every edge e i consists of r consecutive vertices and for anytwo consecutive edges e i and e i +1 it holds | e i ∩ e i +1 | = ℓ (this requires that r − ℓ divides n andthus such ℓ -overlapping Hamilton cycle has n/ ( r − ℓ ) edges). We say that a hypergraph is ℓ -hamiltonian if it contains an ℓ -overlapping Hamilton cycle. Frieze [21] determined the thresholdfor the appearance of 1-overlapping 3-uniform Hamilton cycles to be Θ(ln n/n ) (when 4 | n ) andDudek and Frieze [15] extended the result to higher uniformities (2( r − | n ). The divisibilityrequirement was improved to the optimal one (( r − | n ) by Dudek, Frieze, Loh and Speiss [14],see also Ferber [19]. Subsequently, Dudek and Frieze [16] determined thresholds for general ℓ -overlapping Hamilton cycles and a randomized algorithm to find ( r − ω ( n ℓ − r ) is an asymptotically optimal threshold for ℓ -Hamiltonicity (for ℓ ≥ ω ( f ) denotesany function g such that g ( n ) /f ( n ) → ∞ as n → ∞ . For another type of Hamilton cycles, theso-called Berge
Hamilton cycles, the threshold was determined recently by Poole [35].In the first part of this paper we observe that Riordan’s result [37] can be extended to r -uniform hypergaphs leading to a general theorem about spanning structures in random hy-pergraphs. We will recover results of Dudek and Frieze [16] on ℓ -hamiltonicity (2 ≤ ℓ < r ) andalso discuss thresholds for other spanning structures such as hypercubes, lattices, spheres andpowers of Hamilton cycles in hypergraphs.Let H = ( V, E ) be an r -uniform hypergraph with n vertices. We write v ( H ) for | V | and e ( H ) for | E | . We denote by deg( v ) the degree of a vertex v in H : deg( v ) := |{ e : v ∈ e }| , and∆( H ) is defined to be the maximum vertex degree in H , i.e. ∆( H ) := max v ∈ V deg( v ). Let e H ( v ) = max { e ( F ) : F ⊆ H, v ( F ) = v } , then the following parameter introduced in [37] will beresponsible for the upper bound on the threshold γ ( H ) = max r +1 ≤ v ≤ n (cid:26) e H ( v ) v − (cid:27) . Our first result is the following.
Theorem 1.1.
Let r ≥ be an integer and H = H ( i ) be a sequence of r -uniform hypergraphswith n = n ( i ) vertices, ∆ = ∆( H ) and e ( H ) = α (cid:0) nr (cid:1) = α ( n ) (cid:0) nr (cid:1) edges. Let p = p ( n ) : N → [0 , . f the following conditions are satisfied α (cid:18) nr (cid:19) > nr and p (cid:18) nr (cid:19) → ∞ , (1) and np γ ( H ) ∆ − → ∞ , (2) then a.a.s. the random r -uniform hypergraph H ( r ) ( n, p ) contains a copy of H . We remark, that for r = 2 this was already shown by Riordan in [37, Theorem 2.1] using thesecond moment argument. In fact, the proof for hypergraphs will follow allong the lines of hisoriginal argument, but requires adaptations at various places. We provide the details below inSection 2 and in Section 3 we discuss its applications to some particular spanning hypergraphs.1.2. Universality.
For a family F of r -uniform hypergaphs we say that an r -uniform hyper-graph H is F -universal if every hypergraph F ∈ F occurs as a copy in H . Let F ( r ) ( n, ∆) denotethe family of all r -uniform hypergraphs F with maximum degree ∆( F ) ≤ ∆ on n vertices.In the graph case, the first result concerned almost spanning universality due to Alon, Ca-palbo, Kohayakawa, R¨odl, Ruci´nski and Szemer´edi [6] who showed that for p = C (ln n/n ) / ∆ therandom graph G ( n, p ) is F (2) ((1 − ε ) n, ∆)-universal a.a.s. Then, Dellamonica, Kohayakawa, R¨odland Ruci´nski [13] proved for any given ∆ ≥ G ( n, p ) is F (2) ( n, ∆)-universal a.a.s. providedthat p ≥ C (ln n/n ) / ∆ , where C > d ( H ) = max H ′ ⊆ H { e ( H ) /v ( H ) } of a graph into the statement, Ferber, Nenadov and Peter [20] showed that for the uni-versality for all graphs with maximum degree ∆ and maximum density d the probability p = ω (∆ n − / (2 d ) ln n ) suffices. Thus, for sparser graphs with d < ∆ / n . Very recently, Conlon,Ferber, Nenadov and ˇSkori´c [12], building on [20] proved that for every ε >
0, ∆ ≥ p = ω ( n − / (∆ − ln n ) the random graph G ( n, p ) is F (2) ((1 − ε ) n, ∆)-universal a.a.s. improvingthe almost spanning result from [6].Our second result is on the universality of H ( r ) ( n, p ) for the family F ( r ) ( n, ∆), where we showthat a natural bound on p ≥ C (ln n/n ) / ∆ suffices. Theorem 1.2.
For every r ≥ and any integer ∆ ≥ , there exists a constant C > , suchthat for p = C (ln n/n ) / ∆ the random r -uniform hypergraph H ( r ) ( n, p ) is F ( r ) ( n, ∆) -universala.a.s. In the proof of Theorem 1.2 (see Section 4) we employ a strategy of Dellamonica, Kohayakawa,R¨odl and Ruci´nski [13], but a shortcut will be obtained by using similar notions of good prop-erties that were introduced by Ferber, Nenadov and Peter [20].Another natural problem concerns the existence and explicit constructions of graphs that areuniversal for some family of graphs. For an excellent survey on this problem see Alon [2] and thereferences therein. First nearly optimal universal graphs for F (2) ( n, ∆) (∆ ≥
3) with O ( n ) ver-tices and O ( n − / ∆ ln / ∆ n ) edges were given by Alon, Capalbo, Kohayakawa, R¨odl, Ruci´nskiand Szemer´edi [7]. It was also noted by the same authors in [6] that any such universal graphhas to contain Ω( n − / ∆ ) edges. As mentioned in [4] in the case ∆ = 2 the square of a Hamiltoncycle is F (2) ( n, ∆)-universal (and thus 2 n edges are enough in this case). Subsequently, Alonand Capalbo gave better constructions of F (2) ( n, ∆)-universal graphs (∆ ≥
3) with O ( n ) ver-tices and O ( n − / ∆ ) edges [4] and another one with n vertices and O ( n − / ∆ ln / ∆ n ) edges [5].We exploit their constructions to obtain sparse F ( r ) ( n, ∆)-universal hypergraphs. Proposition 1.3.
For every r ≥ , any integer ∆ ≥ and any F (2) ( n, ∆) -universal graph G there is an r -uniform hypergraph H with V ( H ) = V ( G ) and | E ( H ) | ≤ | E ( G ) | n r − , which is F ( r ) ( n, ∆) -universal. t will follow from Proposition 1.3 that if G is explicitly constructible then so is the hypergraph H . This proposition together with the results of Alon and Capalbo [4, 5] yields for ∆ ≥ F ( r ) ( n, ∆)-universal hypegraphs with O ( n ) vertices and O ( n r − / ∆ ) edges andwith n vertices and O ( n r − / ∆ ln / ∆ n ) edges. Note that the case ∆ = 1 is trivial.Our next proposition shows that, for r ≥ F ( r ) ( n, ∆)-universalhypergraphs. These are obtained from appropriate universal random graphs from [12, 13]. Proposition 1.4.
Let r ≥ and ∆ ≥ be integers. Then there exists an F ( r ) ( n, ∆) -universalhypergraph H with n vertices and Θ (cid:16) n r − r (ln n ) r (cid:17) edges. Moreover, for every ε > there ex-ists an F ( r ) ( n, ∆) -universal hypergraph H with (1 + ε ) n vertices and ω n r − ( r ) ( r − − (ln n ) ( r ) ! edges. These hypergraphs are thus sparser than the random universal hypergraph from Theorem 1.2.On the other hand, any F ( r ) ( n, ∆)-universal hypergraph has to contain Ω( n r − r/ ∆ ) edges andthus the exponent in its density is off by at most the factor of 2. We discuss Propositions 1.3and 1.4 in Section 5.1.3. Notation.
Let H = ( V, E ) be an r -uniform hypergraph. The hypergraph induced by asubset of the vertices W ⊆ V in H is denoted by H [ W ] := (cid:16) W, E ( H ) ∩ (cid:0) Wr (cid:1)(cid:17) . The shadowgraph H ′ is obtained from H by replacing every edge e ∈ E ( H ) by all possible (cid:0) r (cid:1) subsetsof cardinality two (we ignore multiple edges). By K ( r ) n we denote the complete r -uniformhypergraph (cid:0) [ n ] , (cid:0) nr (cid:1)(cid:1) .An alternating sequence of vertices and edges v , e , v , e , . . . , v t , e t , v t +1 is called a path oflength t from v to v t +1 if v i , v i +1 ∈ e i for all i ∈ [ t ]. If there is a path from u to v , then we saythat u and v are connected. This defines an equivalence relation on V . Similarly to the graphcase, we say that a hypergraph H is connected if there is a path between any two vertices of H . A component in an r -uniform hypergraph is a maximally connected subhypergraph. Thedistance between two vertices u and v in H ist the minimal length over all paths from u to v ,and if they are in different components then we set it to ∞ .The neighbourhood N H ( v ) of a vertex v is the set of vertices which are contained in an edgetogether with v N H ( v ) = { w ∈ V \ { v } : ∃ e ∈ E s.t. { w, v } ⊆ e } . For a subset of the vertices W ⊆ V , the neighbourhood in H is N H ( W ) = S w ∈ W N H ( w ). Theset W is called t -independent in a hypergraph H , if the distance between v ∈ W and w ∈ W in H is at least t + 1. A 1-independent set is independent in the usual sense.To simplify readability, we will omit in the calculations floor and ceiling signs whenever theyare not crucial in the arguments.2. Proof of Theorem 1.1
The overall proof strategy of Theorem 1.1 is the same as Riordan’s in [37], which is an elegantsecond moment argument. In fact, a large part of the proof proceeds along the same lines andwe are thus going to use the same notation. We will also provide reference to the relevantpart of [37], especially for lemmas that hold verbatim or by a straightforward modification forhypergraphs, but we also include some of these for the sake of readability. Still, some of thesteps require more effort to be generalized (in particular Lemma 2.3 below) and we provide fulldetails for them. We try to be brief anyway.The proof deals instead of H ( r ) ( n, p ) with the related model H ( n, p (cid:0) nr (cid:1) ), which is the proba-bility space of all labelled r -uniform hypergraphs with the vertex set [ n ] and exactly p (cid:0) nr (cid:1) edgeswith a uniform measure. Thus, for r = 2 this is the standard model G ( n, m ). A corresponding tatement in the model H ( r ) ( n, p ) is then obtained from H ( n, p (cid:0) nr (cid:1) ) by a standard argument,i.e. conditioning on the number of edges in H ( r ) ( n, p ).One considers the random variable X which counts copies of H in H ( n, p (cid:0) nr (cid:1) ) and analyzesthe quantity f := E ( X ) / ( E X ) . It is enough to show that f = 1 + o (1), since then one infersby Chebyshev’s inequality: P [ X = 0] ≤ P [ | X − E X | ≥ E X ] ≤ Var( X ) E ( X ) = f − o (1) . (3)Before we give more details, let us briefly state the steps that are geared towards the estima-tion of f as done in [37], since this is the path we are going to pursue as well:(1) pondering f , it is shown that f ≤ (1 + o (1)) e − − pp α ( nr ) S H , where S H is a sum thatdepends on all subhypergraphs of H , which will be introduced below;(2) then it is shown that S H ≤ (1 + o (1)) e − pp α ( nr ) S ′ H , where S ′ H runs only over certain goodsubhypergraphs of H – this step requires most adaptation and we provide full details inLemma 2.3 below;(3) one can further simplify S ′ H and bound it above by another quantity T ′ H – this is donein Lemma 2.4 and its proof is sketched in the Appendix;(4) in the penultimate step, T ′ H is bounded by e T ′′ H , where T ′′ H is the sum over all goodconnected hypergraphs;(5) finally, it is shown T ′′ H = o (1) (we sketch a proof in Lemma 2.5 that can be found in theAppendix) and combining the estimates, the desired bound on f follows: f ≤ (1 + o (1)) e − − pp α ( nr ) S H ≤ (1 + o (1)) e − − pp α ( nr ) e − pp α ( nr ) S ′ H ≤ (1 + o (1)) T ′ H ≤ (1 + o (1)) e T ′′ H ≤ (1 + o (1)) e o (1) = 1 + o (1) . Before we proceed, let us collect some useful estimates that involve α and p for future refer-ence. Lemma 2.1.
Suppose that the conditions (1) and (2) hold. Then, we have n − r p − ∆ r − → and ∆ = o ( n / ) ,α (cid:18) nr (cid:19) p − = o (1) ,αp − ∆ = o ( n − / ) , p − ∆ n − r = o ( n / ) and p − α (cid:18) nr (cid:19) = o ( n / ) . Proof.
Observe first that α (cid:0) nr (cid:1) > n/r implies γ ( H ) ≥ r − . Since p ≤
1, it follows from (2) that∆ = o ( n / ), and rearranging yields with γ ( H ) ≥ r − that n − r p − ∆ r − → p = ω (cid:16) (∆ /n ) r − (cid:17) . Since α (cid:0) nr (cid:1) ≤ ∆ n/r it follows α ≤ ∆ (cid:0) n − r − (cid:1) − . The combination of the two estimates yields α (cid:0) nr (cid:1) p − = o (1).To obtain the remaining estimates, we combine the lower bound on p = ω (cid:16) (∆ /n ) r − (cid:17) with α ≤ ∆ (cid:0) n − r − (cid:1) − , thus obtaining αp − ∆ ≤ p − ∆ (cid:0) n − r − (cid:1) − = o ( n − / ) and p − α (cid:0) nr (cid:1) = o ( n / ) . (cid:3) The proof starts by writing f as the sum over all 2 e ( H ) subhypergraphs F of H that involve X F ( H ) and X F ( K ( r ) n ), which are the number of subhypergraphs of H (resp. of K ( r ) n ) that areisomorphic to F . The following lemma is a consequence of Lemmas 3.1, 3.2 and 4.1 from [37](these involve only binomial coefficients and thus can be applied verbatim to hypergraphs). emma 2.2. Suppose that α (cid:0) nr (cid:1) , p (cid:0) nr (cid:1) → ∞ and that α (cid:0) nr (cid:1) p − → (as n tends to infinity).Then with c = − pp − α we get f ≤ (1 + o (1)) e − − pp α ( nr ) X F ⊆ H c e ( F ) X F ( H ) X F ( K ( r ) n ) . (4)The third condition above ( α (cid:0) nr (cid:1) p − →
0) holds by Lemma 2.1.Notice that every component in an r -uniform hypergraph has either one vertex (isolatedvertex) or at least r vertices. We define the function r ( F ) := n − k ( F ) − ( r − k r ( F ) where n is the number of vertices in F , k ( F ) is the number of isolated vertices in F and k r ( F ) is thenumber of components in F , which are not isolated vertices. As in [37] one introduces the sum S H = X F ⊆ H ( p − − e ( F ) (1 + n − ) r ( F ) X F ( H ) X F ( K ( r ) n ) . (5)This allows us to get the bound of the form (this is Lemma 4.2 in [37]) f ≤ (1 + o (1)) e − − pp α ( nr ) S H , which follows by estimating ln (cid:16) pp − α (cid:17) ≤ αp , e ( F ) ≤ ∆ r ( F ) and the use of αp − ∆ = o ( n − / )(which follows from Lemma 2.1).Next one would like to estimate S H by the following sum S ′ H = ′ X F ⊆ H ( p − − e ( F ) r ( F ) X F ( H ) X F ( K ( r ) n ) , (6)where P ′ is the sum over subhypergraphs F ⊆ H such that none of the components of F consists of a single (isolated) edge (such hypergraphs are referred to as good in [37]).The following lemma has the same conclusion as Lemma 4.3 from [37]. In the case of r -uniform hypergraphs ( r ≥
3) one needs to be more careful and the estimates are somewhatdifferent from those in [37]. Therefore we provide its full proof below.
Lemma 2.3. If H is any r -uniform hypergraph with maximum degree ∆ and the conditions (1) and (2) hold, then S H ≤ (1 + o (1)) e − pp α ( nr ) S ′ H . Proof.
Let F be some hypergraph from the sum P ′ in S ′ H (such F we call good). Thus, F isan r -uniform hypergraph with v isolated vertices and no isolated edges. We define S ′ [ F ] to bethe contribution to S ′ H that comes from the isomorphism class of a good hypergraph F ⊆ H ,i.e. S ′ [ F ] = ( p − − e ( F ) r ( F ) X F ( H ) X F ( K ( r ) n ) . We write F t for a hypergraph obtained from a good F with v isolated vertices by adding t ≤ v/r isolated edges to it. Let S [ F ] be the contributionof all subhypergraphs of H that are isomorphic to F i for some i , where 0 ≤ i ≤ v/r . Thus, S [ F ] = P v/ri =0 ( p − − e ( F i ) (1 + n − ) r ( F i ) X Fi ( H ) X Fi ( K ( r ) n ) . Every hypergraph from the sum in S H canbe reduced to a good F by deleting all isolated edges. Therefore we have S ′ H = P S ′ [ F ] and S H = P S [ F ], where the sums are over all isomorphism classes of good subhypergraphs of H . To prove the lemma it is thus sufficient to bound S [ F ] /S ′ [ F ] for every good F ⊆ H by(1 + o (1)) e − pp α ( nr ).Let F ⊆ H be a good hypergraph with v isolated vertices, then X F t ( K ( r ) n ) = X F ( K ( r ) n ) · t ! (cid:18) vr (cid:19)(cid:18) v − rr (cid:19) · . . . · (cid:18) v − rt + rr (cid:19) and X F t ( H ) ≤ X F ( H ) · t ! e H ( v ) e H ( v − r ) · . . . · e H ( v − rt + r ) . etting β w = e H ( w ) / (cid:0) wr (cid:1) we obtain X F t ( H ) X F t ( K ( r ) n ) ≤ X F ( H ) X F ( K ( r ) n ) β v β v − r . . . β v − rt + r t ! . Since e ( F t ) = e ( F ) + t and r ( F t ) = r ( F ) + t we have S [ F ] S ′ [ F ] ≤ − r ( F ) (1 + n − ) r ( F ) v/r X t =0 ( p − − t (1 + n − ) t β v β v − r . . . β v − rt + r t ! . (7)Next we take a closer look at the β w terms. Since ∆( H ) ≤ ∆ we can bound e H ( w ) ≤ w ∆ /r and β w ≤ (cid:0) w ∆ r (cid:1) (cid:0) wr (cid:1) − ≤ ∆ r r − w r − . Therefore we estimate the product of β w s as follows t − Y i =0 β v − ir ≤ (cid:0) ∆ r r − (cid:1) t t − Y i =0 ( v − ir ) ! − ( r − ≤ ∆ t (cid:18) ( ⌊ v/r ⌋ − t )! ⌊ v/r ⌋ ! (cid:19) r − . By approximating factorials with Stirling’s formula we obtain t − Y i =0 β v − ir ≤ (cid:18) e r − ∆ ⌊ v/r ⌋ r − (cid:19) t . Thus, we further upper bound S [ F ] /S ′ [ F ] by S [ F ] S ′ [ F ] ≤ − r ( F ) (1 + n − ) r ( F ) v/r X t =0 ( p − − t (1 + n − ) t (cid:18) e r − ∆ ⌊ v/r ⌋ r − (cid:19) t t ! . (8)From Lemma 2.1 it follows that p − ∆ n − r = o ( √ n ) and therefore p − ∆ v − r = o (( n/v ) r − √ n ) (9)and in the following we will distinguish four cases.Suppose 0 ≤ v ≤ n/ (100 r ln n ). Then we use (9) to upper bound each term in the sumfrom (8) by n v = exp( n/ (100 r )). On the other hand we have r ( F ) ≥ n − vr > n/ (2 r ). It followsthat 2 − r ( F ) dominates each of the at most n/r terms in the sum and the factor (1 + n − ) r ( F ) as well. This gives us S [ F ] /S ′ [ F ] = o (1). If v = 0 then we trivially have S [ F ] /S ′ [ F ] = o (1) aswell.Next we assume that n/ (100 r ln n ) < v ≤ n − (ln n ) r − √ n . We can interpret the sum in (8)as the first v/r + 1 terms in the expansion of exp (cid:16) ( p − − n − / ) e r − ∆ ⌊ v/r ⌋ r − (cid:17) , which leadsto S [ F ] S ′ [ F ] ≤ − r ( F ) (1 + n − ) r ( F ) exp (cid:18) p − e r − ∆ ⌊ v/r ⌋ r − (cid:19) . (10)Again we have r ( F ) ≥ n − vr ≥ (ln n ) r − √ nr , whereas exp (cid:16) p − e r − ∆ ⌊ v/r ⌋ r − (cid:17) = exp (cid:0) o ((ln n ) r − √ n ) (cid:1) by (9). Thus, we have S [ F ] /S ′ [ F ] = o (1).Assume now that n − (ln n ) r − √ n < v ≤ n − √ n . Similarly as in the previous case onegets r ( F ) ≥ √ n/r and exp (cid:16) p − e r − ∆ ⌊ v/r ⌋ r − (cid:17) = exp ( o ( √ n )). Again one gets S [ F ] /S ′ [ F ] = o (1) asbefore.Finally, let v > n − √ n and we are going to use the inequality (7) to estimate S [ F ] /S ′ [ F ]. Webound β w with e ( H ) (cid:0) wr (cid:1) − = α (cid:0) nr (cid:1) (cid:0) wr (cid:1) − which is α (cid:0) nr (cid:1) (1 + O ( n − / )) for w ≥ n − ( r + 1) √ n .This gives us √ n X t =0 ( p − − t (1 + n − ) t (cid:0) α (cid:0) nr (cid:1) (1 + O ( n − / )) (cid:1) t t ! ≤ exp (cid:18) − pp α (cid:18) nr (cid:19) (cid:16) O ( n − / ) (cid:17)(cid:19) . y Lemma 2.1 we have − pp α (cid:0) nr (cid:1) n − / = o (1). Thus,exp (cid:18) − pp α (cid:18) nr (cid:19) (cid:16) O ( n − / ) (cid:17)(cid:19) ≤ (1 + o (1)) exp (cid:18) − pp α (cid:18) nr (cid:19)(cid:19) . As for t > √ n , we estimate the rest by (8) and using (9) it follows: v/r X t = √ n ( p − − t (1 + n − ) t (cid:18) e r − ∆ ⌊ v/r ⌋ r − (cid:19) t t ! ≤ v/r X t = √ n o (1) t = o (1) . Combining together we obtain: S [ F ] S ′ [ F ] ≤ (1 + o (1)) e α ( nr ) − pp + o (1) = (1 + o (1)) e α ( nr ) − pp .Therefore, for every good F , we get in any of the four possible cases that S [ F ] S ′ [ F ] ≤ (1 + o (1)) e α ( nr ) − pp . This yields S H /S ′ H ≤ (1 + o (1)) e α ( nr ) − pp completing the proof. (cid:3) So far we have f ≤ (1 + o (1)) e − − pp α ( nr ) S H and S H ≤ (1 + o (1)) e − pp α ( nr ) S ′ H , thus f ≤ (1 + o (1)) S ′ H . In the following one bounds S ′ H by bounding first X F ( H ) /X F ( K ( r ) n ). Lemma 2.4 (an adaptation of Lemma 4.4 from [37]) . Let H be any r -uniform hypergraph withmaximum degree ∆ and F ⊆ H , then X F ( H ) X F ( K ( r ) n ) ≤ (2( r − r ( F ) e r ( F )+( r − k r ( F ) n r ( F )+( r − k r ( F ) The lemma above bounds S ′ H as follows: S ′ H ≤ ′ X F ⊆ H ( p − − e ( F ) (4( r − r ( F ) e r ( F )+( r − k r ( F ) n r ( F )+( r − k r ( F ) =: T ′ H . Proceeding exactly as in [37], one introduces ψ ( F ) := ( p − − e ( F ) (4( r − r ( F ) e r ( F )+( r − kr ( F ) n r ( F )+( r − kr ( F ) which is multiplicative: it holds ψ ( F ∪ F ) = ψ ( F ) ψ ( F ) for any two vertex-disjoint hyper-graphs F and F (i.e. for all e ∈ E ( F ) and f ∈ E ( F ) one has e ∩ f = ∅ ). Since every goodhypergraph consists of maximally connected edge-disjoint good hypergraphs we get T ′ H = ′ X F ⊆ H ψ ( F ) ≤ ∞ X i =1 t ! ′′ X F ⊆ H ψ ( F ) t , where P ′′ is the sum over connected good hypergraphs. We set T ′′ H = P ′′ F ⊆ H ψ ( F ), thus theabove shows T ′ H ≤ e T ′′ H . Lemma 2.5 (an adaptation of Lemma 4.5 [37]) . For every r -uniform hypergraph H on [ n ] wehave T ′′ H ≤ ne r n X s = r +1 (cid:18) r ! ∆ n (cid:19) s − p − e H ( s ) , (11) where ∆ is the maximum degree of H . Now we are in a position to finish the argument. We further estimate T ′′ H using the above asfollows: T ′′ H ≤ ne r n X s = r +1 (cid:18) r ! ∆ n (cid:19) s − p − e H ( s ) ≤ e r r ! n X s = r +1 (cid:16) r ! ∆ p − e H ( s ) / ( s − n − (cid:17) s − . Therefore we get T ′′ H ≤ e r r ! P ns = r +1 (cid:0) r ! ∆ p − γ ( H ) n − (cid:1) s − , which by condition (2) tendsto zero as n goes to infinity. Thus, T ′′ H = o (1), and with T ′ H ≤ e T ′′ H and f ≤ (1 + o (1)) S ′ H ≤ T ′ H e obtain f ≤ o (1) and then by Chebyshev’s inequality (3) the statement of Theorem 1.1follows for H ( n, p (cid:0) nr (cid:1) ). 3. Applications of Theorem 1.1
First we obtain the following two corollaries.
Corollary 3.1.
Let r , ∆ ≥ be integers and H = H ( i ) a sequence of r -uniform hypergraphs with n = n ( i ) vertices, ∆( H ) ≤ ∆ , e ( H ) > n/r and γ ( H ) = e ( H ) / ( n − . Then for p = ω (cid:0) n − /γ ( H ) (cid:1) the random graph H ( r ) ( n, p ) contains a copy of H a.a.s., while for every ε > we have for p ≤ (1 − ε )( e/n ) /γ that P ( H ⊆ H ( r ) ( n, p )) → .Proof. Since ∆ is fixed and γ ( H ) ≤ (1 + o (1))∆, the conditions (1) and (2) are satisfied.Theorem 1.1 yields the first part of the claim.Let X be the number of copies of H in H ( r ) ( n, p ) and we estimate its expectation E ( X ) asfollows: E ( X ) ≤ n ! p e ( H ) ≤ √ n (1 − ε ) e ( H ) ( n/e ) = o (1) . Now Markov’s inequality P ( X ≥ ≤ E ( X ) yields the second part of the corollary. (cid:3) We call a hypergraph
H d -regular if every vertex of H has degree d . Corollary 3.2.
Let r ≥ be an integer and H = H ( i ) be a sequence of ∆ -regular r -uniformhypergraphs where ∆ = ω (ln( n ) − /r ) but ∆ = o ( n / ) . Then for every ε > we have that H ( r ) ( n, p ) contains a.a.s. H if p = (1 + ε ) n − r/ ∆ . Furthermore P ( H ⊆ H ( r ) ( n, p )) → for p ≤ n − r/ ∆ , i.e. p = n − r/ ∆ is a sharp threshold for the appearance of copies of H in H ( r ) ( n, p ) .Proof. Let X count the copies of H in H ( r ) ( n, p ) and for p ≤ n − r/ ∆ we have P ( X ≥ ≤ E ( X ) ≤ n ! n − re ( H ) / ∆ = n ! n − n = o (1) . Next we bound γ ( H ) as follows: ∆ /r ≤ γ ( H ) ≤ ∆ r (∆ / ( r − +1)(∆ / ( r − − . This is obtained from theestimate e H ( v ) ≤ min { ∆ v/r, (cid:0) vr (cid:1) } by considering two cases whether v ≤ ∆ / ( r − + 1 or not.Let ε ∈ (0 ,
1) and notice that (1) is satisfied. It also holds that n (cid:16) (1 + ε ) n − r/ ∆ (cid:17) γ ( H ) ∆ − ≥ (cid:16) (1 + ε ) n /γ ( H ) − r/ ∆ ∆ − r (1+ o (1)) / ∆ (cid:17) γ ( H ) ≥ (cid:16) (1 + ε ) n − r/ (∆ / ( r − ) (1 + o (1)) (cid:17) γ ( H ) → ∞ , and therefore Theorem 1.1 is applicable and the statement follows. (cid:3) Thus Theorem 1.1 (Corollaries 3.1 and 3.2) states that under some technical conditions thethreshold for the appearance of the spanning structure comes from the expectation thresholddefined in the introduction. Further it should be noted that the appearance of 1-overlappingHamilton cycles and also perfect matchings and of general F -factors the structure in questionappears as soon as some local obstruction (isolated vertices, no vertices in some copy of a fixedgraph F ) disappears. Thus, there seem to be two types of behaviour that are responsible forthe threshold for the appearance of a bounded degree spanning structure.In the following we derive asymptotically optimal thresholds for the appearance of variousspanning structures in H ( r ) ( n, p ) which are consequences of the Corollaries 3.1 and 3.2.3.1. Hamilton Cycles.
The following is a slightly weaker version of Dudek and Frieze [16].
Corollary 3.3.
For all integers r > ℓ ≥ , ( r − ℓ ) | n and p = ω ( n ℓ − r ) the random hypergraph H ( r ) ( n, p ) is ℓ -hamiltonian a.a.s. roof. Denote by C ( r,ℓ ) an ℓ -overlapping Hamilton cycle on n vertices. It is not difficult to seethat γ ( C ( r,ℓ ) ) = n ( r − ℓ )( n − . Indeed, let V ⊆ [ n ] be a set of size v < n . Then C ( r,ℓ ) [ V ] is aunion of vertex-disjoint ℓ -overlapping paths, where an ℓ -overlapping path of length s consists of s ( r − ℓ ) + ℓ ordered vertices and edges are consecutive segments intersecting in ℓ vertices. Thisgives: e ( C ( r,ℓ ) [ V ]) ≤ ( v − ℓ ) / ( r − ℓ ) and from v − ℓ ( r − ℓ )( v − ≤ n ( r − ℓ )( n − we get γ ( C ( r,ℓ ) ) = n ( r − ℓ )( n − .Since e ( C ( r,ℓ ) ) > n/r , ∆( C ( r,ℓ ) ) = ⌈ rr − ℓ ⌉ and n r − ℓ ) /n →
1, Corollary 3.1 implies the state-ment. (cid:3)
Cube-hypergraphs.
The r -uniform d -dimensional cube-hypergraph Q ( r ) ( d ) was studiedin [11] and its vertex set is V := [ r ] d and its hyperedges are r -sets of the vertex set V that alldiffer in one coordinate. Thus, Q ( r ) ( d ) has r d vertices, dr d − edges and is d -regular. In the case r = 2 this is the usual definition of the (graph) hypercube. The following corollary is a directconsequence of Corollary 3.2. Corollary 3.4.
For all integers r ≥ , ε > and p = r − r + ε it holds P ( Q ( r ) ( d ) ⊆ H ( r ) ( r d , p )) tends to as d tends to infinity. On the other hand, P ( Q ( r ) ( d ) ⊆ H ( r ) ( r d , r − r )) → as d → ∞ . We remark that, in the case r = 2, Riordan [37] proved even better dependence of ε on d ,and similar dependence can be shown for r > Lattices.
Another example considered in [37] was the graph of the lattice L k , whose vertexset is [ k ] and two vertices are adjacent if their Euclidean distance is one. There it is shownthat p = n − / is asymptotically the threshold. One can view L k as the cubes Q (2) (2) (theseare cycles C ) glued ‘along’ the edges. We define the ℓ -overlapping hyperlattice L ( r ) ( ℓ, k ) asthe r -uniform hypergraph where we glue together ( k − copies of Q ( r ) (2) that overlap on ℓ hyperedges accordingly. Thus, L (2) (1 , k ) is just the usual graph lattice L k . Corollary 3.5.
Let r ≥ and k be an integer. For p = ω (cid:0) n − / (cid:1) (where n = ( k − r ) ) therandom r -uniform hypergraph H ( r ) ( n, p ) contains a copy of L ( r ) ( r − , k ) a.a.s. Moreover, for p = n − / , P ( L ( r ) ( r − , k ) ⊆ H ( r ) ( n, p )) → as k (and thus n ) tends to infinity.Proof. Observe that L := L ( r ) ( r − , k ) has ( k − r ) vertices (which can be associated with[ k − r ] ) and 2( k − k − r ) edges.We aim to show that e L ( v ) ≤ v − r ) for all v ≥ r + 1. We argue similarly as in [37].Observe that e L ( v ) ≤ v = r + 1. Let now L ′ be an arbitrary subhypergraph of L on v + 1 ≤ ( k − r ) vertices such that e ( L ′ ) = e L ( v + 1). It is easy to see that there is avertex of degree 2 in L ′ (take ( i, j ) such that ( i + 1 , j ), ( i, j + 1) V ( L ′ )). It follows that then e L ( v + 1) ≤ e L ( v ) + 2 for v > r + 1 giving e L ( v ) ≤ v − r ) for all v ≥ r + 1.It follows that γ ( L ) ≤ np γ = ω (1) yields the first part.Markov’s inequality yields the second part. (cid:3) Spheres.
Let r ≥ G be a planar graph on n vertices with a drawing all of whosefaces are cycles of length r . We define a sphere S rn as an r -uniform hypergraph all of whose edgescorrespond to the faces of that particular drawing (note that a sphere is not unique). Observethat we get from Euler’s formula for planar graphs the condition 2 v ( S rn ) − r − e ( S rn ). Corollary 3.6.
Let r ≥ and S be some sphere S rn with ∆ = ∆( S rn ) . Then for p = ω (cid:0) ∆ r − n − ( r − / (cid:1) the random r -uniform hypergraph H ( r ) ( n, p ) contains a copy of S a.a.s.Proof. From Euler’s formula it follows that e S ( v ) ≤ v − r − and therefore γ ( S ) = 2 / ( r − r -edges in this induced hypergraph we immediatelyget γ = 2 / ( r − (cid:3) .5. Powers of ( r − -overlapping Hamilton cycles. Consider an ( r − C ( r,r − with n vertices which are ordered cyclically. Given an integer i , we definean i -th power C ( r ) ( i ) of C ( r,r − to consist of all r -tuples e such that the maximum distance inthis cyclic ordering between any two vertices in e is at most r + i −
2. In the graph case, thethreshold for the appearance of C (2) ( i ) follows from Riordan’s result [37] for i ≥ i = 2 an approximate threshold due to K¨uhn and Osthus [32] is known. If wecount the edges of C ( r ) ( i ) by their leftmost vertex we get e ( C ( r ) ( i )) = n (cid:0) r + i − r − (cid:1) . Theorem 3.7.
Let r ≥ and i ≥ be integers. Suppose that p = ω ( n − / ( r + i − r − )) , then therandom hypergraph H ( r ) ( n, p ) contains a.a.s. a copy of C ( r ) ( i ) . This threshold is asymptoticallyoptimal.Proof. One can argue similarly to Proposition 8.2 in [32] to show γ ( C ( r ) ( i )) ≤ (cid:0) r + i − r − (cid:1) + O r,i (1 /n ). The statement follows from Theorem 1.1. We omit the details. (cid:3) Proof of Theorem 1.2
Outline.
Our proof follows a similar strategy as the one of Dellamonica, Kohayakawa,R¨odl and Ruci´nski [13] for universality in random graphs, but to verify good properties of arandom hypergraph necessary for embeddings we combine it with the recent approach of Ferber,Nenadov and Peter [20] who studied random graphs as well.We will embed any bounded degree hypergraph F ∈ F ( r ) ( n, ∆) into the random hypergraph H = H ( r ) ( n, p ) with p = C (ln n/n ) / ∆ in stages. For this we will partition most of the verticesof F into 3-independent sets X ,. . . , X t (this is achieved by coloring greedily the third powerof the shadow graph of F ) and the remaining vertices will form the set of linear size N F ( X t ),where the hypergraphs F [ N F ( x )] and the link of x in F look the same for all x ∈ X t . Therandom hypergraph H is then prepared as follows: the vertex set of H is partitioned into sets V , V ,. . . , V t where “most” of the vertices lie in V . The property that we use first is that onecan embed into H [ V ] the induced hypergraph F [ N F ( X t )] for any F ∈ F ( r ) ( n, ∆) so that therestrictions on future images for ∪ i ∈ [ t ] X i still offer many choices. In later rounds, we embedeach X i into available vertices from V ∪ S j ≤ i V j by Hall’s condition ( X i s are 3-independent).In order to verify Hall’s condition for small subsets of X i the sets V i will assist us in this.4.2. Proof of Theorem 1.2.
Let H = ( V, E ) be an r -uniform hypergraph. The link of v in H is a subset of (cid:0) Vr − (cid:1) consisting of all ( r − v link H ( v ) = (cid:26) e ′ ∈ (cid:18) Vr − (cid:19) : e ′ ∪ { v } ∈ E (cid:27) . For a hypergraph H and a vertex v we define its profile P H ( v ) in H as follows P H ( v ) = ( N H ( v ) , E ( H [ N H ( v )]) , link H ( v ))and say that two profiles P H ( v ) and P H ( v ) are equivalent ( P H ( v ) ∼ = P H ( v )) if there is an iso-morphism ϕ that takes H [ N H ( v )] to H [ N H ( v )] and ( N H ( v ) , link H ( v )) to ( N H ( v ) , link H ( v )).We call N H ( v ) the vertices of the profile.Let P ( r ) (∆) be the set of all possible profiles ( Z, E , E ) that we encounter for any F ∈∪ n ∈ N F ( r ) ( n, ∆) (up to equivalence). Then any | Z | ≤ ( r − Z, E ) is an r -uniform hy-pergraph with maximum degree ∆ − Z, E ) is an ( r − | P ( r ) (∆) | by a functionexponential is some polynomial in ∆, but since ∆ is a constant, all we will care about is that | P ( r ) (∆) | is a constant as well that depends on ∆ only.The following lemma prepares any F ∈ F ( r ) ( n, ∆) for future embedding into H ( r ) ( n, p ). Lemma 4.1.
Let r ≥ and ∆ ≥ be integers. Then for t = r ∆ , any ε ≤ | P ( r ) (∆) | − ( t − − and any F ∈ F ( r ) ( n, ∆) there exists a partition of V ( F ) in X ∪ · · · ∪ X t (where some X i s mightbe empty) with the following conditions: | X t | = εn , X = N H ( X t ) , (2) every x ∈ X t has the same profile P F ( x ) (up to equivalence), (3) and X i is -independent for i = 1 , . . . , t .Proof. Let F ∈ F ( r ) ( n, ∆) be given and G be its shadow graph. The third power G of G is the graph which we obtain by connecting any pair of vertices of distance at most 3 by anedge. We estimate the maximum degree of G as follows: ∆( G ) ≤ ( r − ∆ . Clearly, G is ( t − Y ,. . . , Y t − be the color sets in some coloring of V ( G ) such that | Y | ≤ | Y | ≤ . . . ≤ | Y t − | . The sets Y i are 3-independent in F as well because the shadow of apath of length 3 in F contains a path of length 3 in G , which gives an edge in G in contradictionto the coloring above.We can choose a subset X t ⊆ Y t − of size εn ≤ n | P ( r ) (∆) | − ( t − − of vertices with thesame profile in F (up to equivalence). We set X := N F ( X t ) and define X i = Y i \ X for i = 1 , . . . , t − X t − = Y t − \ ( X ∪ X t ). The partition V ( F ) = X ∪ · · · ∪ X t satisfies therequired conditions. (cid:3) Given a partition of V ( F ) from the above lemma, it follows from properties (1)–(3) that F [ X ]is the disjoint union of εn copies of the same r -uniform hypergraph isomorphic to F [ N F ( x )] forall x ∈ X t . Furthermore, the third condition implies that any edge e ∈ E ( H ) intersects each X i in at most one vertex for i = 1,. . . , t .Let H = ( V, E ) be an r -uniform hypergraph. Let L be a family of pairwise disjoint k -subsetsof (cid:0) Vr − (cid:1) and we write V ( L ) for ∪ e ∈ L : L ∈L e . For a subset W ⊆ V \ V ( L ) we define the auxiliarybipartite graph B ( H, L , W ) with the vertex classes L and W , where L ∈ L and w ∈ W forman edge if and only if L ⊆ link H ( w ). Roughly speaking, for every unembedded x ∈ V ( F ) theset L = L x ∈ L will consist of the images of the already fully embedded ( r − F ( x ) and the following definition which resembles the one of good graphs from [20] providesessential properties that will assist us while embedding F into H ( r ) ( n, p ). Definition 4.2.
We say that an r -uniform hypergraph H is ( n, r, p, t, ε, ∆) -good if there exists apartition V ( H ) = V ∪ V ∪ · · · ∪ V t , where | V i | = εn/ (10 t ) for i = 1 , . . . , t , and | V | = (1 − ε/ n that satisfies the following conditions: (1) For any profile ( Z, E , E ) ∈ P ( r ) (∆) there exists a family F of εn vertex-disjoint copiesof the profile ( Z, E , E ) with vertices in V and edges E present in H . This familyinduces a family F of pairwise disjoint copies of E in (cid:0) V r − (cid:1) . Furthermore, for any W ⊆ V ( H ) \ V ( F ) with | W | ≤ ( p/ − ∆ / | N B ( H, F ,W ) ( W ) | ≥ ( p/ ∆ | W | εn/ holds. (2) Let ≤ k ≤ ∆ and L be any collection of disjoint k -subsets of (cid:0) V ( H ) r − (cid:1) . If |L| ≤ ( p/ − k / , then for any i = 1 , . . . , t with V ( L ) ∩ V i = ∅ we have | N B ( H, L ,V i ) ( L ) | ≥ ( p/ k |L| | V i | / . (3) Let ≤ k ≤ ∆ and L be any collection of disjoint k -subsets of (cid:0) V ( H ) r − (cid:1) . If |L| ≥ C ′ ( p/ − k ln n , then for any W ⊆ V ( H ) \ V ( L ) with | W | ≥ C ′ ( p/ − k ln n the graph B ( H, L , W ) has at least one edge, where C ′ = k ( r −
1) + 2 . The following two lemmas establish the connection between H ( r ) ( n, p ), good hypergraphs and F ( r ) ( n, ∆)-universality. Lemma 4.3.
For integers r ≥ , ∆ ≥ , t ≥ and ε ≤ / ( r ∆) , there exists a C > such thatfor p ≥ C (ln n/n ) / ∆ the random hypergraph H ( r ) ( n, p ) is ( n, r, p, t, ε, ∆) -good a.a.s. Lemma 4.4.
For integers r ≥ , ∆ ≥ and ε ≤ | P ( r ) (∆) | − r − ∆ − , there exists a C > suchthat for p ≥ C (ln n/n ) / ∆ , every (cid:0) n, r, p, r ∆ , ε, ∆ (cid:1) -good hypergraph is F ( r ) ( n, ∆) -universal. he proof of Theorem 1.2 follows immediately from Lemmas 4.3 and 4.4. Thus, it remainsto prove both lemmas.We will make use of the following version of Chernoff’s inequality, see e.g. [23, Theorem 2.8]. Theorem 4.5 (Chernoff’s inequality) . Let X be the sum of independent binomial randomvariables, then for any δ ≥ P [ X ≤ (1 − δ ) E ( X )] ≤ exp( − δ E ( X ) / . For an r -uniform hypergraph F with v ( F ) ≥ r , we define d (1) ( F ) := e ( F ) v ( F ) − and we also set d (1) ( F ) = 0 if v ( F ) ≤ r −
1. Now we set m (1) ( F ) := max (cid:8) d (1) ( F ′ ) : F ′ ⊆ F (cid:9) . We will use thefollowing theorem that deals with almost spanning factors in random (hyper-)graphs. Theorem 4.6. [Theorem 4.9 in [23] ] For every r -uniform hypergraph F and every ε < /v ( F ) there is a C > such that for p ≥ Cn − /m (1) ( F ) , the random hypergraph H ( r ) ( n, p ) contains εn vertex-disjoint copies of F a.a.s.Proof of Lemma 4.3. Let r , ∆, t and ε ≤ / ( r ∆) be given, furthermore we assume that p ≥ C (ln n/n ) / ∆ , where C is a sufficiently large constant that depends only on ε , r , ∆ and t . Wewill not specify C explicitly but it will be clear from the context how it should be chosen.We expose H ( r ) ( n, p ) in two rounds and write H ( r ) ( n, p ) = H ( r ) ( n, p ) ∪ H ( r ) ( n, p ), where p = p ≥ p/ − p ) = (1 − p )(1 − p ). In the first round we will find thefamiles F and in the second round we show that properties (1)-(3) of Definition 4.2 all holdwith probability 1 − o (1). In the beginning we arbitrarily partition V into V ∪ V ∪ · · · ∪ V t suchthat | V | = n − εn/
10 and V i = εn/ (10 t ) for i = 1 , . . . , t . st round. For a given profile (
Z, E , E ) ∈ P ( r ) (∆) we have that the maximum degree of G = ( Z, E ) is at most ∆ −
1. We estimate m (1) ( G ) ≤ max s ≥ r (∆ − sr ( s − ≤ ∆ −
1. Theorem 4.6implies that there exist εn vertex-disjoint copies of G in H ( r ) ( n, p ) all of whose vertices arecontained inside V a.a.s. Indeed, we apply Theorem 4.6 to H ( r ) ( | V | , p ) where | V | ≥ (1 − ε/ n > ( r − εn + εn/
10. We denote this family by F ( Z,E ) .Since there are constantly many (at most | P ( r ) (∆) | ) r -uniform hypergraphs G on at most( r − −
1, we will find simultaneously εn vertex-disjointcopies of any such G a.a.s. within V . Therefore, with a given profile ( Z, E , E ) ∈ P ( r ) (∆), weassociate a family F of εn vertex-disjoint copies ( Y, E ′ , E ′′ ) with ( Y, E ′ ) ∈ F ( Z,E ) and such that( Y, E ′ , E ′′ ) ∼ = ( Z, E , E ). This gives us a family F of the E ′′ s for such a profile, thus showingthe first part of the property (1) of Definition 4.2. nd round. From now on we work in H = H ( r ) ( n, p ).Fix some profile ( Z, E , E ) ∈ P ( r ) (∆) and the corresponding family F found in the firstround. The family F induces a family F of disjoint copies of E in (cid:0) V r − (cid:1) . Let W be a subsetof V ( H ) \ V ( F ) with | W | ≤ ( p/ − ∆ /
2. For every L ∈ F let X L be the random variablewith X L = 1 if and only if L ⊆ link H ( w ) for some w ∈ W . This gives us | N B ( H , F ,W ) ( W ) | = P L ∈F X L . The X L are independent and since P [ xL ∈ E ( B ( H , F , W ))] ≥ ( p/ ∆ , we compute P [ X L = 0] ≤ (1 − ( p/ ∆ ) | W | ≤ − | W | ( p/ ∆ + | W | ( p/ ≤ − | W | ( p/ ∆ / . From this we obtain E X L ∈F X L ≥ ( p/ ∆ | W ||F | / |F | = εn ≥ ε ( C/ ∆ | W | (ln n ) / δ = 1 / P X L ∈F X L < ( p/ ∆ | W | |F | / ≤ exp( − ε ( C/ ∆ | W | (ln n ) /
16) = n − ε ( C/ ∆ | W | / . (12) ince there are at most n s choices for a set W of size s we can bound, for C large enough, theprobability that there is a set W violating property (1) for F by o (1).The number of different profiles in P ( r ) (∆) depends only on ∆ and thus also the numberof F s. Thus taking the union bound over the probability that there is a set W violating ourcondition for some family F is still o (1). This verifies property (1) of Definition 4.2.To verify properties (2) and (3) of Definition 4.2, we use the edges of H ( r ) ( n, p ). Let k ∈ [∆], L be a collection of disjoint k -subsets of (cid:0) Vr − (cid:1) with |L| ≤ ( p/ − k / i ∈ [ t ] such that V ( L ) ∩ V i = ∅ . For v ∈ V i , let X v be the random variable with X v = 1 if and only if L ⊆ link H ( v ) for some L ∈ L . Thus | N B ( H , L ,V i )) ( L ) | = P v ∈ V i X v . As above we obtain P [ X v = 0] ≤ (cid:16) − ( p/ k (cid:17) |L| ≤ − |L| ( p/ k + |L| ( p/ k ≤ − |L| ( p/ k / . We have E X v ∈ V i X v ≥ ( p/ k |L|| V i | / | V i | = εn t ≥ ε ( C/ k |L| (ln n ) / (20 t )and from Chernoff’s inequality with δ = 1 / P X v ∈ V i X v ≤ ( p/ k |L|| V i | / ≤ exp( − ( p/ k |L|| V i | / ≤ n − ε ( C/ k |L| / (320 t ) . There are less than n ( r − k |L| possibilities to choose L . Therefore, for C large enough, theprobability that there exists k ∈ [∆] and sets L and V i that violate property (2) of Definition 4.2is o (1).Finally, we verify that property (3) holds a.a.s. in H . For this we set ℓ = C ′ ( p/ − k ln n andlet k ∈ [∆]. It suffices to consider only sets L and W ⊆ V \ V ( L ) each of size ℓ . For two suchsets L and W the probability that an edge in B ( H , L , W ) is present equals ( p/ k and thereforethe probability that there are no edges is (1 − ( p/ k ) ℓ ≤ exp( − ℓ ( p/ k ).There are less than n ( r − kℓ choices for L and less than n ℓ choices for W . Thus we can boundthe probability that there are sets L and W of size ℓ violating property (3) byexp[(( r − kℓ + ℓ ) ln n − ℓ ( p/ k ] = exp[(( r − k + 1 − C ′ ) ℓ ln n ] = o (1) . (cid:3) Proof of Lemma 4.4.
Let r , ∆, ε ≤ | P ( r ) (∆) | − r − ∆ − be given and let C . r , ∆, t := r ∆ and ε . Furthermore we assume that p ≥ C (ln n/n ) / ∆ , where C is a sufficiently large constant that depends only on ε , r , ∆ and C .
3. We will not specify C explicitly but it will be clear from the context how it should bechosen.Let H be an ( n, r, p, t, ε, ∆)-good hypergraph and fix the partition V ∪ V ∪ · · · ∪ V t of V ( G ) asspecified by Definition 4.2. Fix any F ∈ F ( r ) ( n, ∆) and apply Lemma 4.1 with r , ∆, t = r ∆ and ε to obtain a partition of V ( F ) in X ∪ · · · ∪ X t with the properties (1)–(3) from Lemma 4.1.An embedding of F into G is an injective map φ : V ( F ) → V ( H ), where edges are mappedonto edges. We start with constructing an embedding φ that X maps into V ⊂ V ( H ). FromLemma 4.1, property (2), we know that every x ∈ X t has the same profile in F . Therefore,let ( Z, E , E ) be the profile of any x ∈ X t . By Definition 4.2, property (1), there is a family F of copies of ( Z, E , E ) with vertices in V . Since F [ X ] is the disjoint union of εn copies of( Z, E ) we can construct φ by mapping bijectively every copy ( N F ( x ) , F [ N F ( x )] , link F ( x )) toone member ( Y, E ′ , E ′′ ) of F . This is for sure a valid embedding of F [ X ] into H .Now we construct φ i from φ i − for i = 1 , . . . , t by embedding X i such that φ i (cid:16) F [ ∪ ij =0 X j ] (cid:17) ⊆ H . The available vertices for this step are V ∗ i = ( V ∪ · · · ∪ V i ) \ Im( φ i − ). For x ∈ X i we collect he images of the already fully embedded ( r − F ( x ) in L ( x ) := ( φ i − ( e ) : e ∈ link F ( x ) ∩ (cid:18)S i − j =0 X j r − (cid:19)) . Since X i is 3-independent we have L ( x ) ∩ L ( x ) = ∅ for x , x ∈ X i and we set L i = { L ( x ) : x ∈ X i } is a collection of vertex-disjoint sets in (cid:0) V ( H ) r − (cid:1) . A possible image for x ∈ X i is any v ∈ V ∗ i , for which L ( x ) ⊆ link H ( v ). It remains to find an L i -matching in B i = B ( H, L i , V ∗ i )since then we set φ i ( x ) := v for every edge vL ( x ) in this matching and, since any edge e ∈ E ( F )intersects X i in at most one vertex, we obtain φ i (cid:16) F [ ∪ ij =0 X j ] (cid:17) ⊆ H .To guarantee an L i -matching in B i we will verify Hall’s condition. Let U ⊆ L i and one needsto show that | N B i ( U ) | ≥ | U | holds. We assume ∅ 6∈ U , because otherwise N B i ( U ) = V ∗ i and | V ∗ i | ≥ |L i | .First we verify Hall’s condition for all sets U with | U | ≤ | V ∗ i | − εn/
10. Notice that thereexists a k ∈ [∆] and a subset U ′ of size at least | U | / ∆ and | L | = k for every L ∈ U ′ . If | U ′ | ≤ ( p/ − k /
2, then by property (2) of Definition 4.2 we have for C large enough | N B i ( U ) | ≥ | N B i ( U ′ ) | ≥ ( p/ k | U ′ || V i | / ≥ ε ( C/ k | U | (ln n ) / (40 t ∆) ≥ | U | . If ( p/ − k / < | U ′ | < C ′ ( p/ − k ln n , then we take any subset U ′′ of size ( p/ − k / C large enough | N B i ( U ) | ≥ | N B i ( U ′′ ) | ≥ ( p/ k | U ′′ || V i | / ≥ ε ( C/ k | U | / (20 C ′ t ∆) ≥ | U | . If | U ′ | > C ′ ( p/ − k ln n , then | U | > C ′ ( p/ − k ln n and there are no edges between U and V ∗ i \ N B i ( U ) in B i . Therefore, property (3) of Definition 4.2 yields for C large enough | V ∗ i \ N B i ( U ) | < C ′ ( p/ − k ln n ≤ C ′ ( C/ − k ( n/ ln n ) k/ ∆ ln n ≤ εn/ , and thus | N B i ( U ′ ) | > | V ∗ i | − εn/
10 which verifies Hall’s condition in B i for | U | ≤ | V ∗ i | − εn .For i = 1 , . . . , t − | S ti =1 V i | = εn/
10 and | X t | = εn , that | V ∗ i | − | X i | ≥ ( n − | Im φ i − | − εn/ − ( n − | Im φ i − | − εn ) ≥ / εn and therefore |L i | = | X i | ≤ | V ∗ i | − εn/ L i -matchings in B i for i ∈ [ t −
1] one after each other extending at eachstep our embedding.In the last step, we have | V ∗ t | = | X t | = εn and, by the partitioning of V ( F ) with X = N F ( X t )we have L t = F , where F is the family guaranteed by property (1) of Definition 4.2. Since wealready saw that | N B t ( U ) | ≥ | U | for all U ⊆ L t with | U | ≤ |L t | − εn/
10 in B t = B ( H, L t , V ∗ t ), itsuffices to verify | N B t ( W ) | ≥ | W | for all W ⊆ V ∗ i with | W | ≤ εn/
10. If | W | > ( p/ − ∆ / W ′ ⊆ W of size exactly ( p/ − ∆ / W ′ := W .By property (1) of Definition 4.2, we have | N B t ( W ′ ) | ≥ ( p/ ∆ | W ′ | εn/ , which is at least εn/ > εn/ ≥ | W | if W ′ ( W and is at least ε ( C/ ∆ (ln n ) | W ′ | / > | W | if W = W ′ . Therefore, N B t ( U ) ≥ | U | for all | U | ≥ |L t | − εn/
10 as well and there exists a (perfect) L t -matching in B t that allows us to finish embedding F into H . (cid:3) In the proof of Theorem 1.2 we only considered the case of constant ∆. Similarly to thearguments in [20] this can be extended to the range when ∆ is some function of n but then thiswould affect the lower bound on the probability p . Furthermore, the proof yields a randomizedpolynomial time algorithm that on input H ( r ) ( n, p ) embeds a.a.s. any given F ∈ F ( r ) ( n, ∆) into H ( r ) ( n, p ). All steps of the proof can be performed in polynomial time and the only place wherewe need to use additional random bits is to split H ( r ) ( n, p ) into H ( r ) ( n, p ) ∪ H ( r ) ( n, p ) andthis can be done similarly as was done in [1]. . Sparse universal hypergraphs
First we observe that any F ( r ) ( n, ∆)-universal r -uniform hypergraph must possess Ω( n r − r/ ∆ )edges. Indeed, it follows e.g. from a result of Dudek, Frieze, Ruci´nski and ˇSileikis [17] that forany ∆ ≥ r ≥ r -uniform ∆-regular hypergraphs on n vertices(whenever r | n ∆) is Θ (cid:16) (∆ n )!(∆ n/r )!( r !) ∆ n/r (∆!) n (cid:17) . Thus, the number of non-isomorphic r -uniform ∆-regular hypergraphs on n vertices is Ω (cid:16) (∆ n )!(∆ n/r )!( r !) ∆ n/r (∆!) n n ! (cid:17) and a similar bound holds for thecardinality of F ( r ) ( n, ∆). On the other hand an r -uniform hypergraph with m edges containsexactly (cid:0) mn ∆ /r (cid:1) hypergraphs with n ∆ /r edges. Thus, it holds (cid:0) mn ∆ /r (cid:1) = Ω (cid:16) (∆ n )!(∆ n/r )!( r !) ∆ n/r (∆!) n n ! (cid:17) and solving for m yields m = Ω (cid:0) n r − r/ ∆ (cid:1) .The random hypergraph H ( r ) ( n, p ) with p = C (ln n/n ) / ∆ is F ( r ) ( n, ∆)-universal (by Theo-rem 1.2) and has Θ( n r − / ∆ (ln n ) / ∆ ) edges and thus the exponent in the density of H ( r ) ( n, p ) isoff by roughly a factor of r from the lower bound Ω (cid:0) n − r/ ∆ (cid:1) on the density for any F ( r ) ( n, ∆)-universal hypergraph. In the following we show how one can construct sparser F ( r ) ( n, ∆)-universal r -uniform hypergraphs out of the universal graphs from [4, 5].For a given graph G we define the r -uniform hypergraph H r ( G ) on the vertex set V ( G ) and E ( H r ( G )) = { f ∈ (cid:0) V ( G ) r (cid:1) : ∃ e ∈ E ( G ) with e ⊆ f } . Given a hypergraph H and a graph G , wesay that G hits H if for all f ∈ E ( H ) there is an e ∈ E ( G ) with e ⊆ f . We define σ ( H ) to bethe smallest maximum degree ∆( G ) over all graphs G that hit H . Lemma 5.1.
For F ∈ F ( r ) ( n, ∆) we have σ ( F ) ≤ ∆ .Proof. Suppose there is an F ∈ F ( r ) ( n, ∆) with σ ( F ) > ∆ and let G be an edge-minimal graphthat hits H and with ∆( G ) = σ ( H ) > ∆. Take a vertex v of degree at least ∆ + 1 in G . If wedelete any edge e ∈ E ( G ) that is incident to v there will be a hyperedge f ∈ E ( F ) with e ⊆ f so that f does not contain any edge from E ( G ) \ { e } , by the edge-minimality of G . Thus, everyedge e ∈ E ( G ) incident to v corresponds to a different hyperedge f in E ( F ) with e ⊆ f . Sincedeg( v ) ≥ ∆ + 1 there must be at least ∆ + 1 hyperedges in F incident to v , a contradiction. (cid:3) Lemma 5.2.
Let r ≥ and ∆ ≥ be integers and G be a F (2) ( n, ∆) -universal graph. Then H r ( G ) is universal for the family of r -uniform hypergraphs F with σ ( F ) ≤ ∆ .Proof. Let F be an r -uniform hypergraph with σ ( F ) ≤ ∆ and n vertices. By definition of σ ( F )there exists a graph G ′ with n vertices that hits F and G ′ ∈ F (2) ( n, ∆), and thus G ′ can beembedded into G . Since G ′ hits H , the hypergraph H r ( G ) is defined in such a way that anyembedding of G ′ into G is an embedding of F into H r ( G ). (cid:3) Lemmas 5.1 and 5.2 together imply Proposition 1.3 that yields constructions of sparse F ( r ) ( n, ∆)-universal hypergraphs based on universal graphs from [4, 5].Next we turn to yet another possibility to construct F ( r ) ( n, ∆)-universal hypergraphs out ofuniversal graphs. For a given graph G we define an r -uniform hypergraph K r ( G ) on the vertexset V ( G ) with hyperedges being the vertex sets of the copies of K r in G . Lemma 5.3.
Let ∆ ≥ and r ≥ be integers. If G is a F (2) ( n, ( r − -universal graph then K r ( G ) is an F ( r ) ( n, ∆) -universal hypergraph. In particular, K r ( G ) has at most e ( G )∆( G ) r − edges.Proof. We notice first that the shadow graph F ′ of any F ∈ F ( r ) ( n, ∆) has n vertices andmaximum degree at most ( r − F ′ ∈ F (2) ( n, ( r − ϕ be an embedding of F ′ into G . But then, by definition of K r ( G ), ϕ is also an embedding of F into K r ( G ).Furthermore, we can extend any edge of G to a clique in at most ∆( G ) r − many ways.Therefore, e ( K r ( G )) ≤ e ( G )∆( G ) r − . (cid:3) The universal graphs constructed in [4, 5] have m edges and maximum degree O ( m/n ).This yields by Proposition 1.4 constructions of F ( r ) ( n, ∆)-universal hypergraphs which are ust some polylog factors away from the constructions obtained by applying Proposition 1.3.However, if we take an appropriate random graph G = G ( n, p ) then we get a better bound than e ( G )∆( G ) r − on the cliques K r which leads to even sparser universal hypergraphs for r ≥ Proof of Proposition 1.4.
To obtain the first part of the statement, we take G to be the randomgraph G ( n, p ) with p = C (ln n/n ) / (( r − . By the result of Dellamonica, Kohayakawa, R¨odland Ruci´nski [13], G is F (2) ( n, ( r − K r in the random graph is n − / ( r − and that the numberof cliques K r in G ( n, p ) for p ≫ n − / ( r − is Θ( n r p ( r )) a.a.s. Thus, the number of edges in H := K r ( G ) is therefore Θ( n r p ( r )) = Θ (cid:16) n r − r (ln n ) r (cid:17) . Now Lemma 5.3 implies that H is F ( r ) ( n, ∆)-universal.As for the second part, so let ε > G to be the random graph G ((1 + ε ) n, p )with p = ω ( n − / (( r − − ln n ). By the result of Conlon, Ferber, Nenadov and ˇSkori´c [12], G is F (2) ( n, ( r − G has Θ( n r p ( r )) many cliques K r a.a.s. This yields an F ( r ) ( n, ∆)-universal hypergraph H := K r ( G ) with (1 + ε ) n vertices and ω n r − ( r ) ( r − − (ln n ) ( r ) ! edges. (cid:3) Concluding remarks
We believe that it is possible to further reduce the number of edges towards the lower boundΩ( n r − r/ ∆ ) by adjusting the constructions in [4, 5, 7] to hypergraphs. In particular, in viewof Lemma 5.2 it would be interesting to know whether it is true that σ ( F ) ≤ ⌈ /r ⌉ for all F ∈ F ( r ) ( n, ∆), which would lead to almost optimal universal r -uniform hypergraphs for all r ≥
3. For F ∈ F ( r ) ( n, ∆) it is easy to obtain a hitting graph G with at most n ∆ /r edges andthus of average degree at most 2∆ /r , but the maximum degree could be ( r − ⌈ /r ⌉ . We leave it to future work.The bound on p = Ω (cid:0) (ln n/n ) / ∆ (cid:1) in Theorem 1.2 is presumably not optimal and it comesfrom the fact that any ∆-set of (cid:0) [ n ] r − (cid:1) is expected to lie in roughly p ∆ n = Ω(ln n ) many edges(a.a.s. by Chernoff’s inequality) and thus, some reasonable natural embedding should alwayssucceed. As for the lower bound, so if (cid:0) t − r − (cid:1) ≤ ∆ and t divides n , then by a theorem of Johansson,Kahn and Vu [25], the factor of K ( r ) t (which is a collection of n/t vertex-disjoint copies of K ( r ) t )is a member of F ( r ) ( n, ∆). Since the threshold probability [25] for the appearance of such factoris Θ (cid:16) (ln n ) / ( tr ) n − ( t − / ( tr ) (cid:17) and in view of n − ( t − / ( tr ) < n − / ( t − r − ), the best lower bound on p we are aware of is Ω (cid:16) (ln n ) / ( tr ) n − ( t − / ( tr ) (cid:17) .As already mentioned in the introduction, Ferber, Nenadov and Peter studied in [20] univer-sality of G ( n, p ) for the class of graphs F on n vertices with maximum degree at most ∆ and themaximal density d . They showed that the random graph G ( n, p ) with p = ω (∆ n − / (2 d ) ln n )is universal for this family. The recent almost spanning result of Conlon, Ferber, Nenadov andˇSkori´c [12] builds on [20]. We believe that similar improvements can be obtained for random r -uniform hypergraphs ( r ≥ References [1] P. Allen, J. B¨ottcher, Y. Kohayakawa, and Y. Person,
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The only difference in the proofs of Lemmas below to their graph counterparts in [37] is thatwe work with the shadow graph of a hypergraph.
Sketch of the proof of Lemma 2.4.
This lemma is a straightforward adaptation of Lemma 4.4from [37]. Let Y F ( H ) be the number of labelled copies of F in H . One observes that then Y F ( H ) /Y F ( K ( r ) n ) = X F ( H ) /X F ( K ( r ) n ) holds. Since Y F ( K ( r ) n ) = n !, one needs to estimate Y F ( H ). We will embed first exactly one vertex from each of the k r ( F ) nontrivial components.This can be done in ( n ) k r ( F ) ways. Next we can embed ( r −
1) vertices of each componentby embedding one particular edge. This can be done in at most ∆( r − H . Thisgives at most (∆( r − k r ( F ) possibilities in total. Finally, all the remaining r ( F ) − k r ( F )vertices from the nontrivial components can be embedded in at most (( r − r ( F ) − k r ( F ) ways. The isolated vertices can be embedded in at most k ( F )! ways. We estimate Y F ( H ) ≤ ( n ) k r ( F ) (∆( r − k r ( F ) (( r − r ( F ) − k r ( F ) k ( F )!. We obtain: Y F ( H ) Y F ( K ( r ) n ) ≤ ( n ) k r ( F ) (∆( r − k r ( F ) (( r − r ( F ) − k r ( F ) k ( F )! n ! ≤ (2( r − r ( F ) e r ( F )+( r − k r ( F ) n r ( F )+( r − k r ( F ) . (cid:3) Sketch of the proof of Lemma 2.5.
The proof is similar to the proof of Lemma 4.5 [37]. Onerewrites T ′′ H by going over all good connected hypergraphs F on s vertices (then r ( F ) = s − ( r − k r ( F ) = 1) and upper bounds the sum as follows: T ′′ H ≤ n X s = r +1 (4( r − s − r +1 e s − n s − X V e H ( s ) X m =0 (cid:18) e H ( s ) m (cid:19) ( p − − m ≤ e r − n X s = r +1 (12 r !∆) s − r +1 n s − X V p − e H ( s ) , where the second sum is over all s -element sets V such that H [ V ] is connected.We consider the shadow graph H ′ of H [ V ]. Now every V ⊆ [ n ] as above also induces asubgraph in H ′ which is connected and therefore contains a spanning tree. We can estimate thenumber of V s by estimating the number of labelled trees in H ′ on s vertices and then unlabellingthese.Given a labelled tree G on s vertices there are at most n (∆( r − r − s − r ways ofmapping it into H ′ : n accounts for the first vertex of G , then (in H ) we can choose next ( r − r − r −
1) wayssince ∆( H ′ ) ≤ ∆( r − n (∆( r − r − s − r s s − trees and unlabellinggives us at most n (∆( r − r − s − r s s − /s ! ≤ n (∆ r !) s − r +1 e s sets V . This implies T ′′ H ≤ ne r n X s = r +1 (cid:18) r ! ∆ n (cid:19) s − p − e H ( s ) . (cid:3) oethe-Universit¨at, Institut f¨ur Mathematik, Robert-Mayer-Str. 10, 60325 Frankfurt amMain, Germany E-mail address : parczyk | [email protected] | [email protected]