Proofs of auxiliary lemmas

6.4 Proofs of auxiliary lemmas

Proof of Lemma 6.3

Letr,∆,tandε ≤1/(r∆)be given, furthermore we assume thatp≥ C(logn/n)^1/∆, whereCis a sufficiently large constant that depends only onε,r,∆andt. We will not specifyCexplicitly but it will be clear from the context how it should be chosen.

We expose H^(r)(n, p)in two rounds and writeH^(r)(n, p) = H^(r)(n, p1)∪ H^(r)(n, p2), wherep1 = p₂≥p/2such that(1−p) = (1−p₁)(1−p₂). In the first round we will find the familiesFand in the second round we show thatP1–P3of Definition 6.2 all hold with probability at least1−o(1). In the beginning we arbitrarily partitionV intoV0∪V1∪ · · · ∪Vtsuch that|V0|=n−εn/10andVi=εn/(10t) fori= 1, . . . , t.

1^stround.. For a given profile(Z, E₁, E₂)∈P^(r)(∆)we have that the maximum degree ofG= (Z, E₁) is at most∆−1. We estimatem1(G)≤max_s≥r^(∆−1)s_r(s−1) ≤∆−1. Theorem 2.19 implies that there exist εnvertex-disjoint copies ofGinH^(r)(n, p₁)all of whose vertices are contained insideV₀a.a.s. Indeed, we apply Theorem 2.19 toH^(r)(|V0|, p)where|V0| ≥ (1−ε/10)n > (r−1)∆εn+εn/10. We denote this family byF_(Z,E1).

Since there are constantly many (at most|P^(r)(∆)|)r-uniform hypergraphsGon at most(r−1)∆

vertices with maximum degree∆−1, we will find simultaneouslyεnvertex-disjoint copies of any suchGa.a.s. withinV0. Therefore, with a given profile(Z, E1, E2)∈P^(r)(∆), we associate a familyF ofεnvertex-disjoint copies(Y, E⁰, E⁰⁰)with(Y, E⁰)∈ F_(Z,E1)and such that(Y, E⁰, E⁰⁰)∼= (Z, E₁, E₂). This gives us a familyF2of copies ofE⁰⁰for this kind of profile, thus showing the first part ofP1from Definition 6.2.

2^ndround.. From now on we work inH=H^(r)(n, p2). Fix some profile(Z, E1, E2)∈P^(r)(∆)and the corresponding familyFfound in the first round. The familyFinduces a familyF2of disjoint copies ofE2in _r−1^V0

. LetW be a subset ofV(H)\V(F2)with|W| ≤(p/2)^−∆/2. For everyL ∈ F2letXL

be the random variable withX_L = 1if and only ifL ⊆ link_H(w)for somew ∈ W. This gives us

|N_B(H,F2,W)(W)|=P

L∈F2XL. TheXLare independent and sinceP[xL∈E(B(H,F2, W))]≥p^∆₂ ≥ (p/2)^∆, we compute using|W| ≤(p/2)^−∆/2

P[XL= 0]≤(1−(p/2)^∆)^|W|≤1− |W|(p/2)^∆+|W|²(p/2)^2∆≤1− |W|(p/2)^∆/2.

From this we obtain

E

"

X

L∈F2

XL

#

≥(p/2)^∆|W||F2|/2^|F2≥^|=εnε(C/2)^∆|W|(logn)/2

and using Chernoff’s inequality, Theorem 2.16, withγ= 1/2we get

P

"

X

L∈F2

XL <(p/2)^∆|W| |F2|/4

#

≤exp(−ε(C/2)^∆|W|(logn)/16) =n^{−ε(C/2)∆|W|/16}. (6.1)

70

6. Universality in random hypergraphs

Since there are at mostn^schoices for a setW of sizeswe can bound, forClarge enough, the proba-bility that there is a setW violatingP1forF2byo(1).

The number of different profiles inP^(r)(∆)depends only on∆and thus also the number of families F2. Thus taking the union bound over the probability that there is a setW violating our condition for some familyF₂is stillo(1). This verifiesP1of Definition 6.2.

To verifyP2andP3, we use the edges ofH^(r)(n, p2). Letk ∈ [∆],Lbe a collection of disjoint

and from Chernoff’s inequality, Theorem 2.16, withγ= 1/2we get

P

There are less thann^(r−1)k|L|possibilities to chooseL. Therefore, forClarge enough, the probability that there existsk∈[∆]and setsLandVithat violateP2iso(1).

Finally, we verify thatP3holds a.a.s. inH. For this we set`= C<sup>0</sup>(p/2)<sup>−k</sup>lognand letk ∈[∆]. It suffices to consider only setsLandW ⊆ V \V(L)each of size`. For two such setsLandW the probability that an edge inB(H,L, W)is present equals p^k₂ ≥ (p/2)^k and therefore the probability that there are no edges is at most(1−(p/2)^k)<sup>`2</sup> ≤exp(−`²(p/2)^k).

There are less thann^{(r−1)k`</sup>choices forLand less thann<sup>`}choices forW. Thus, we can bound the probability that there are setsLandW of size`violatingP3by

exp constant that depends only onε,r,∆andC6.3. We will not specifyCexplicitly but it will be clear from the context how it should be chosen.

Let H be an(n, r, p, t, ε,∆)-good hypergraph and fix the partitionV0∪V1 ∪ · · · ∪Vtof V(G) as specified by Definition 6.2. Fix anyF ∈ F^(r)(n,∆)and apply Lemma 6.1 withr,∆,t=r³∆³andεto obtain a partition ofV(F)inX₀∪ · · · ∪X_twithQ1–Q3from Lemma 6.1.

6.4 Proofs of auxiliary lemmas

An embedding ofF intoGis an injective mapφ: V(F) → V(H), where edges are mapped onto edges. We start with constructing an embedding φ0 that maps X0 into V0 ⊂ V(H). From Q2, we know that everyx ∈ X_thas the same profile in F. Therefore, let(Z, E₁, E₂)be the profile of anyx ∈ Xt. ByP1, there is a familyF of copies of(Z, E1, E2)with vertices inV0. Furthermore, F[X₀]is the disjoint union ofεncopies of(Z, E₁)which holds because of the3-independence ofX_t (refprop:indep of Lemma 6.1). Therefore, we can constructφ0 by mapping bijectively every copy (NF(x), F[NF(x)],linkF(x))to one member(Y, E⁰, E⁰⁰)ofF. This is for sure a valid embedding of F[X₀]intoH.

Now we constructφifromφ_i−1 fori = 1, . . . , tby embeddingXisuch thatφi F[∪ⁱ_j=0Xj]

⊆H. The available vertices for this step areV_i^∗= (V₀∪ · · · ∪V_i)\Im(φ_i−1). Forx∈X_iwe collect the images of the already fully embedded(r−1)-sets from thelinkF(x)in

L(x) :=

(

φ_i−1(e) :e∈linkF(x)∩ Si−1

j=0Xj

r−1 )

.

SinceX_iis3-independent we haveL(x₁)∩L(x₂) =∅forx₁, x₂∈X_iand we setL_i={L(x) :x∈X_i} which is a collection of vertex-disjoint sets in ^V_r−1^(H)

. A possible image forx∈Xiis anyv ∈V_i^∗, for whichL(x) ⊆ linkH(v). It remains to find anLi-matching inBi = B(H,Li, V_i^∗)since then we set φ_i(x) := v for every edgevL(x)in this matching and, since any edgee ∈ E(F)intersectsX_i in at most one vertex, we obtainφi F[∪ⁱ_j=0Xj]

⊆H.

To guarantee anL_i-matching inB_i we will verify Hall’s condition. LetU ⊆ L_iand one needs to show that|NBi(U)| ≥ |U|holds. We assume∅ 6∈U, because otherwiseNBi(U) =V_i^∗and|V_i^∗| ≥ |Li|.

First, we verify Hall’s condition for all setsU with|U| ≤ |V_i^∗| −εn/10. Notice that there exists a k∈[∆]and a subsetU⁰ ⊆U of size at least|U|/∆and|L|=kfor everyL∈U⁰. If|U⁰| ≤(p/2)^−k/2, then byP2we have forClarge enough

|NB_i(U)| ≥ |NB_i(U⁰)| ≥(p/2)^k|U⁰||Vi|/4≥ε(C/2)^k|U|(logn)/(40t∆)≥ |U|.

If(p/2)^−k/2<|U⁰|< C⁰(p/2)^−klogn, then we take any subsetU⁰⁰ ⊆U⁰of size(p/2)^−k/2and use againP2and|U⁰⁰| ≥2|U|/(C⁰∆ logn)to obtain forClarge enough

|N_Bi(U)| ≥ |N_Bi(U⁰⁰)| ≥(p/2)^k|U⁰⁰||V_i|/4≥ε(C/2)^k|U|/(20C⁰t∆)≥ |U|.

If|U⁰| > C⁰(p/2)^−klogn, then|U| > C⁰(p/2)^−klognand there are no edges betweenU andV_i^∗\ NB_i(U)inBi. Therefore,P3yields forClarge enough

|V_i^∗\NB_i(U)|< C⁰(p/2)^−klogn≤C⁰(C/2)^−k(n/logn)^k/∆logn≤εn/10, and thus|NB_i(U⁰)|>|V_i^∗| −εn/10which verifies Hall’s condition inBifor|U| ≤ |V_i^∗| −εn.

Fori= 1, . . . , t−1it follows from|St

i=1V_i|=εn/10and|X_t|=εn, that

|V_i^∗| − |X_i| ≥(n− |Imφ_i−1| −εn/10)−(n− |Imφ_i−1| −εn)≥9/10εn and therefore|Li|=|Xi| ≤ |V_i^∗| −εn/10.

72

6. Universality in random hypergraphs

Therefore, we findLi-matchings inBifori∈[t−1]one after each other extending at each step our embedding.

In the last step, we have|V_t^∗|=|Xt|=εnand, by the partitioning ofV(F)withX₀=N_F(X_t)we haveLt= F2, whereF2is the family guaranteed byP1. Since we already saw that|NB_t(U)| ≥ |U| for allU ⊆ L_twith|U| ≤ |L_t| −εn/10inB_t=B(H,L_t, V_t^∗), it suffices to verify|N_Bt(W)| ≥ |W|for allW ⊆V_t^∗with|W| ≤εn/10. If|W|>(p/2)^−∆/2then we take an arbitrary subsetW⁰ ⊆W of size exactly(p/2)^−∆/2and otherwise we setW⁰ :=W. ByP1, we have

|NB_t(W⁰)| ≥(p/2)^∆|W⁰|εn/4,

which is at least εn/8 > εn/10 ≥ |W| if W⁰ ( W and is at least ε(C/2)^∆(logn)|W⁰|/4 > |W| if W = W⁰. Therefore, NB_t(U) ≥ |U|for all|U| ≥ |Lt| −εn/10 as well and there exists a (perfect) Lt-matching inBtthat allows us to finish embeddingFintoH.

In the proofs of Lemmas 6.3 and 6.4 (and thus of Theorem 2.9) we only considered the case of con-stant∆. Similarly to the arguments in [56] this also works in the range where∆is some function ofnbut then theC in the bound on the probability is no longer a constant and rather growing ex-ponentially with∆. Furthermore, the proof yields a randomised polynomial time algorithm that on inputH^(r)(n, p)embeds a.a.s. any givenF ∈ F^(r)(n,∆)intoH^(r)(n, p). All steps of the proof can be performed in polynomial time and the only place where we need to use additional random bits is to splitH^(r)(n, p)intoH^(r)(n, p1)∪ H^(r)(n, p2)and this can be done similarly as was done in [4].

Chapter 7

Constructions of universal hypergraphs

In this last chapter we finally give the proofs³⁴of Theorem 2.11, 2.13 and 2.14 aquired with Hetterich and Person [64]. The third theorem is the hardest of these three and we need to prove a new decom-position result for hypergraphs with maximum degree2(cf. Lemma 7.5 and 7.7). For all proofs we need the concept of hitting graphs first introduced in [94], which we will refine in the next section following [64]. But before we come to that let us briefly justify our lower bound on the number of edges in anF^(r)(n,∆)-universal hypergraph as shown in [94]³⁵.

As claimed above we first observe thatanyF^(r)(n,∆)-universalr-uniform hypergraph must pos-sessΩ(n^r−r/∆)edges. Indeed, it follows e.g. from a result of Dudek, Frieze, Ruci ´nski, and ˇSileikis [46]

that for any∆ ≥ 1and r ≥ 3 the number of labelledr-uniform∆-regular hypergraphs on n ver-tices (whenever r|n∆) isΘ _(∆n)!

(∆n/r)!(r!)^∆n/r(∆!)ⁿ

. Thus, the number of non-isomorphic r-uniform

∆-regular hypergraphs onnvertices isΩ _(∆n)!

(∆n/r)!(r!)^∆n/r(∆!)ⁿn!

and a similar bound holds for the cardinality ofF^(r)(n,∆). On the other hand anr-uniform hypergraph withmedges contains exactly

m n∆/r

hypergraphs withn∆/redges. Thus, it holds _n∆/r^m

= Ω _(∆n)!

(∆n/r)!(r!)^∆n/r(∆!)ⁿn!

and solving formyieldsm= Ω n^r−r/∆

.

The random hypergraphH^(r)(n, p)withp=C(logn/n)^1/∆isF^(r)(n,∆)-universal (by Theorem 2.9) and hasΘ(n^r−1/∆(logn)^1/∆)edges and thus the exponent in the density ofH^(r)(n, p)is off by roughly a factor ofrfrom the lower boundΩ n^−r/∆

on the density for anyF^(r)(n,∆)-universal hypergraph.

In the following we show how one can construct sparserF^(r)(n,∆)-universalr-uniform hypergraphs out of the universal graphs from [9, 10].

Im Dokument Spanning structures in random graphs and hypergraphs (Seite 78-83)