Proof for ∆ = 2 - Spanning structures in random graphs and hypergraphs

A similar calculation yieldsF^(r)(n,∆)-universal hypergraphs onnvertices with O(nr−(r+1)/d(r+1)∆/relog2(r+1)/d(r+1)∆/re

edges, which we obtain fromF(n,d(r+ 1)∆/re)-universal graphsGonnvertices with O(n2−2/d(r+1)∆/relog4/d(r+1)∆/ren)

edges.

In contrary to theF chosen as a matching plusP₃we could work with any forestF. To find hitting graphs of small maximum degree we can use similar matching techniques and counting arguments, but in general it is not clear how low we can get. For example, ifFis the pathP_ronrvertices one can show thatF(n,d2(r−1)∆/re)hitsF^(r)(n,∆)onF. This leads to anF^(r)(n,∆)-universal hypergraph onO(n)vertices with O(nr−2(r−1)/d2(r−1)∆/re)edges. It depends on the values of rand∆, which bound is better, but one does not get anything significantly better thanO nr−(r+1)/d(r+1)∆/re

edges and therefore we do not further pursue this here.

Reducing the number of vertices

Note that it is possible to reduce the number of vertices fromO(n)to(1 +ε)nin Theorems 2.10, 2.13, and 2.14 and in Corollary 2.12, for any fixedε > 0, by using a concentrator as was done in [12].

Consider theF^(r)(n,∆)-universal hypergraphH onO(n)vertices and withmedges. Aconcentrator is a bipartite graphC on the vertex setsV(H) and Q, where |Q| = (1 +ε)n such that for every S ⊆V(H)with|S| ≤nwe have|N(S)| ≥ |S|and every vertex fromV(H)hasO_ε(1)neighbours in C. We define a new hypergraphH⁰onQby taking all setsf⁰ ∈ ^Q_r

as edges for which there exists a perfect matching inCfrom an edgef ∈E(H)tof⁰. Since every vertex fromV(H)hasOε(1)degree inC, the hypergraphH⁰hasO_ε(m)edges. It is also not difficult to see thatH⁰isF^(r)(n,∆)-universal.

Indeed, letF ∈ F^(r)(n,∆) and let ϕ: V(F) → V(H)be its embedding intoH. By the property of the concentratorC, there is a matching of ϕ(V(F))inC which we can describe by an injection ψ: ϕ(V(F))→V(H⁰). But now, by construction ofH⁰,ψ◦ϕis an embedding ofF intoH⁰.

7.3 Proof for ∆ = 2

At this point in all cases wherer is not even and r does not divide∆ we do not have construc-tions of F^(r)(n,∆)-universal hypergraphs that match the lower bound Ω(n^r−r/∆) on the number of edges. In this section we will deal with thesmallest open case∆ = 2by constructing optimal F^(r)(n,2)-universal hypergraphs onO(n)vertices withO(n^r/2)edges. So, for example, ifr= 3then Theorem 2.13 yieldsF⁽³⁾(n,2)-universal hypergraphs onO(n)vertices withO(n^3−4/d8/3e) =O(n^5/3) edges, while the lower bound isΩ(n^3/2).

We will first deal with the caser= 3and∆ = 2and then reduce the case of generalrand∆ = 2to this one. Let us say a few words how an improvement fromO(n^5/3)toO(n^3/2)can be accomplished.

We will use the concept of a graphGthat hits some hypergraphH onP3 (the path on3vertices).

If we would follow the arguments in the previous section, then we see that taking a hypergraph 78

7. Constructions of universal hypergraphs

H ∈ F⁽³⁾(n,2)and replacing every hyperedge byP3we can obtain a hitting graphGof maximum degree3and of average degree8/3. Thus, if we would like to use Theorem 2.10 we need to consider F(n,3)-universal graphs, which results in the loss of somen^1/6-factor in the edge density. Instead, we will seek to decompose the hitting graphGinto appropriate subgraphsG1,G2,G3andG4such that every edge ofGlies inexactlythree of the graphsG_i. A decomposition result of Alon and Capalbo from [9] will assist us in this. Finally, following closely the arguments again due to Alon and Capalbo but now from [10] will allow us to construct a universal graphGonO(n)vertices and with maximum degreeO(n^1/4)for a carefully chosen familyF⁰of graphs allowing a decomposition as above, which hitsF⁽³⁾(n,2)onP3. Lemma 7.1 implies then thatHP₃,3(G)isF⁽³⁾(n,2)-universal and hasO(n^3/2) edges.

A graph decomposition result

The following notation is from [9]. LetGbe a graph andS ⊆V(G)be a subset of its vertices. A graph G⁰which is obtained fromGby adding additionally|S|new vertices toGand placing an (arbitrary) matching between these new vertices and the vertices fromSis called anaugmentationofG. We call a graphthinif every of its components is an augmentation of a path or a cycle, or if they contain at most two vertices of degree3. We also call any subgraph of a thin graph thin.

The following decomposition theorem may be seen as a generalisation of Petersen’s Theorem to graphs of odd degree.

Theorem 7.3(Theorem 3.1 from [9]). Let∆be an integer andGa graph with maximum degree∆. Then there are∆spanning subgraphsG1, . . . , G∆such that eachGiis thin and every edge ofGappears in precisely two graphsG_i.

Its proof is built on the Gallai-Edmonds decomposition theorem and is implied by the following lemma.

Lemma 7.4(Lemma 3.3 from [9]). Let∆≥3be an odd integer andGa∆-regular graph. ThenGcontains a spanning subgraph in which every vertex has degree2or 3and every connected component has at most2 vertices of degree3.

We will use the two results above to prove the existence of a hitting graphGwith nice properties so that we can later take advantage of them when constructing a universal graph for the family of such nicehitting graphs.

Lemma 7.5. LetH ∈ F⁽³⁾(n,2). Then there exists a graphGthat hitsHonP3with the following properties:

(i) there are spanning subgraphsG₁,G₂,G₃andG₄ofGsuch that everyG_iis an augmentation of a thin graph, and

(ii) every edge lies inexactlythree of theGi.

Proof. LetH ∈ F⁽³⁾(n,2). We assume first thatHis linear, i.e. edges are always intersecting in at most one vertex. Further, we assume thatHis2-regular³⁶.

36Otherwise we adddummyvertices and edges and obtain a2-regular hypergraph, and, once the desired graphGis con-structed, we delete these dummy vertices fromG.

7.3 Proof for∆ = 2

The rough outline of the proof is to find a graphGthat hitsH onP3 and such thatGcontains a matchingMso thatG\M is an augmentation of a thin graph and if we contract the matching edges fromM inGwe obtain a graph of maximum degree at most3. Decomposing such contracted graph via Theorem 7.3 into thin graphsG⁰₁,G⁰₂andG⁰₃and thenrecontractingedges yields the desired family G₁,. . . ,G₄(whereG₄=G\M).

LetH^∗be the line graph ofH, that isV(H^∗) =E(H)ande6=f ∈E(H)form an edgeef inH^∗if e∩f 6=∅. Thus,H^∗ is a3-regular graph on2n/3vertices. Lemma 7.4 asserts then the existence of a matchingM^∗inH^∗such that inH^∗\M^∗every component has at most2vertices of degree3and all other vertices have degree2. Such a decomposition implies thus that every component ofH^∗\M^∗is either a cycle, or has exactly two vertices, sayaandb, of degree3, so that either there are3internally vertex-disjoint paths betweenaandbor there is one path betweenaandband, additionally,aandb lie on vertex-disjoint cycles (which also do not contain inner vertices from the path betweenaandb).

We assume thataandbare not adjacent, because otherwise we could add the edgeabtoM^∗, splitting this component into two cycles.

From the matchingM^∗we define a subsetD :={v:e∩f ={v}whereef ∈E(M^∗)}. SinceM^∗is a matching in the line graph ofH it follows that no two vertices fromDlie in an edge fromH.

We denote byH_D the hypergraph which we obtain fromH if we delete from the edges ofH the vertices inDbut we keep the edges, obtaining thus a hypergraph on the vertex setV(H)\D, whose edges have cardinality2 or 3. Thus, ifef is an edge inH^∗ and e∩f = {v} then the deletion of vfrom eandf implies that the edgese\ {v}and f \ {v} are no longer adjacent in the line graph (HD)^∗, which corresponds to the deletion of the edgeef inH^∗. This implies that every component ofH^∗\M^∗corresponds to a component ofH_D, and therefore in every component ofH_Dthere are at most two edges of cardinality3and all other edges have cardinality exactly2. Again, the structure of every component ofH_Dis thus either a (graph) cycle, or there are exactly two edges, saygandh, of cardinality3, withg∩h=∅and there are three vertex-disjoint (graph) paths that connect the vertices fromg∪h.

Finally, we come to the definition of the hitting graphG. For every componentCofHD, letDC

be the vertices that have been deleted from the hyperedges inHthat lie now inH_D. Thus, there is a (natural) mapψCbetween the edges fromCof cardinality2andDC:ψC(f) =vif{v} ∪f ∈E(H).

Note that this map is not necessarily injective. Since every vertex fromDlies in exactly two edges of H, it will suffice to explain how we replace the3-uniform edges ofH_Dand the edges ofH incident withDby pathsP3. IfCis the (graph) cycle, then we replace every edge of the form{v} ∪f, where ψ_C(f) = v, byP₃so that the graphG_C obtained contains all the edges fromE(C)and is such that

∆(GC)≤3and the vertices fromDChave degree at most2inGC. IfCcontains exactly two3-uniform edges (sayg andh), then it is possible to replace the edgesg,hand every edge of the form{v} ∪f, whereψC(f) = v, by P3 such that the graphGC satisfies the following: It contains all 2-uniform edges ofC, is such that∆(GC)≤3, the vertices fromDChave degree at most2inGCandGC\DC

is connected and has exactly two vertices of degree3³⁷. The graphGis then the union of allG_C and observe thatGCandGC⁰ intersect inDC∩DC⁰forC6=C⁰and in particular have no common edges.

Furthermore, every vertex fromDhas degree2inG, since it is an image ofψ_Cprecisely twice.

LetM be a matching inGthat saturatesD. Such a matching exists sinceDis independent inG(no

37This is easily done by considering the structure of the componentsCfromH_Ddescribed in the previous paragraph.

7. Constructions of universal hypergraphs

two vertices fromDlie in an edge fromH), every vertex ofDis connected to a vertex of degree2in G\Danddeg(G)≤3. By the definition ofGabove, every component inG\M is an augmentation of a graph with at most two vertices of degree3, and thus an augmentation of a thin graph. We set G4:=G\M. Next, we contract the edges ofM inGobtaining the graphG/M. SinceM saturatesD, which are vertices of degree2inG, it follows thatG/Mhas maximum degree at most3. Theorem 7.3 yields a decomposition ofG/M into thin graphsG⁰₁,G⁰₂,G⁰₃such that every edge ofG/Mappears in precisely two of the graphs. Now we reverse the recontraction procedure. This leads to three graphs G₁,G₂ andG₃ where every edge ofG\M appears in exactly two of the graphs, every edge from M appears in all three of them, and each of theG1,G2andG3is an augmentation of a thin graph.

Together with the graph G₄ = G\M we thus constructed the desired decomposition of a hitting graphG.

If H is not linear, then things get in some sense even easier, so we shall be brief. We proceed essentially in the same way. That is, we define the line graphH^∗ofH, which is now not necessarily 3-regular, but whose maximum degree is at most3. Again, Lemma 7.4 asserts then the existence of a matchingM^∗inH^∗such that inH^∗\M^∗every component has at most2vertices of degree3and all other vertices have degree at most2. We then define the setDas before but in the case that the edge ef ∈M^∗with, say,e={a, b, c}andf ={b, c, d}we simply replace the edgeeby{a, b}andf by{c, d}

without putting anything intoD. Once the components ofHDare identified and the graphsGCare defined we add the edgebc(which we call nonlinear) to those graphsG_C, which contain eitherborc (or both). Then we choose edges into the matchingM as before and add all nonlinear edges such as bctoM. The rest of the argument remains the same.

Recall that the`-th power of a graphG, denoted byG^`, is the graph onV(G), whose vertices at distance at most`inGare connected. It is not difficult to see that a thin graph onnvertices can be embedded intoP_n⁴, and thus, an augmentation of a thin graph intoP_n⁸. This motivates the following general definition.

Definition 7.6((k, r, `)-decomposable graphs). Letk,rand`be integers. A graphGonnvertices is called (k, r, `)-decomposableif there exist kgraphsGiwith the following properties. Every edge ofGappears in exactlyrof theGi and there are mapsgi: Gi → [n], which are injective homomorphisms fromGi intoP_n^`. Then we denote byFk,r,`(n)the family of(k, r, `)-decomposable graphs onnvertices.

We can restate our Lemma 7.5 in the following slightly weaker form.

Lemma 7.7. The familyF4,3,8(n)hitsF⁽³⁾(n,2)on a pathP3.

This lemma implies that it is the familyF4,3,8(n)for which a universal graph is needed. This graph will be constructed in the section below and briefly explained why a desired embedding works, which will follow from the results of Alon and Capalbo from [10].

Constructions of universal graphs

First we briefly describe the construction from [10] ofF(n, k)-universal graphs onO(n)vertices with O(n^2−2/k)edges. One choosesm= 20n^1/k, a fixedd >720and a graphRto be ad-regular graph on mvertices with the absolute value of all but the largest eigenvalues at mostλ(such graphs are called

7.3 Proof for∆ = 2

(n, d, λ)-graphs). One can assume thatλ ≤ 2√

d−1(then R is called Ramanujan) andgirth(R) ≥

3logm/log(d−1). Explicit constructions of such Ramanujan graphs have been found first ford−1 being a prime congruent to1mod 4in [85, 88]. Finally, the graphG_k,nis defined on the vertex set V(R)^kwhere two vertices(x1, . . . , xk)and(y1, . . . , yk)are adjacent if and only if there are at least two indicesisuch thatx_i andy_i are within distance4inR. It is easily seen that such a graphG_k,nhas O(n)vertices,O(n^2−2/k)edges and maximum degreeO(n^1−2/k).

The first step in the proof ofF(n, k)-universality ofG_k,nis Theorem 7.3 implying that any graph F with∆(F) ≤ kis(k,2,4)-decomposable. In what follows we summarise a straightforward gen-eralisation of the central claim from [10] (which is inequality (3.1) there), from which an existence of embedding of any graphG ∈ F(n, k)intoGk,nfollows. Its proof can be taken almost verbatim from [10].

Lemma 7.8. Letk≥3,rand`be natural numbers. For any choice ofkpermutationsgi: [n]→[n]there are khomomorphismsfi: [n] →V(R)from the pathPnto the Ramanujan graphRintroduced above such that the mapf: [n]→V(G_k,r,`(n))defined byf(v) = (f₁(g₁(v)), . . . , f_k(g_k(v)))is injective.

More precisely, thefi’s are inductively constructed as non-returning walks preserving the property that for anyiverticesv₁, . . . , v_i∈V(G),i≤k, one has

|{v∈[n] :f₁(g₁(v)) =v₁, . . . , f_i(g_i(v)) =v_i}| ≤n^(k−i)/k. For the last stepi=kthis is equivalent to injectivity.

Finally, we explain, how we obtainF_k,r,`(n)-universal graphs. The choice of the Ramanujan graph Ralong with the parametersmanddremains the same. The graphGk,r,`(n)is defined on the vertex setV(R)^k and two vertices(x1, . . . , xk)and(y1, . . . , yk)are adjacent if and only if there are at least rindicesisuch thatx_i andy_iare within distance`inR. It is then an easy calculation to show that Gk,r,`(n)hasO(n)vertices, at mostn ^k_r

d^r`m^k−r=O(n^2−r/k)edges and maximum degreeO(n^1−r/k), where the constants in theO-notation depend onk,r,`andd. Lemma 7.8 implies then the following.

Theorem 7.9. Letk≥3,rand`be natural numbers. The graphGk,r,`(n)isFk,r,`(n)-universal.

Proof. LetGbe a(k, r, `)-decomposable graph onnvertices together with a decompositionG₁, . . . , G_k and an injective homomorphismsgi: V(Gi)→[n]fromGiintoP_n^`. Lemma 7.8 asserts the existence of the homomorphismsfi: [n] → V(R)fromPn toRfor everyi∈ [k], so that the mapf: V(G) → V(Gk,r,`(n))given byf(v) = (f1(g1(v)), . . . , fk(gk(v)))is injective.

It is clear that the composition off_iwithg_iis a homomorphism fromG_itoR^`. Furthermore, every edge{u, v}fromGlies inrgraphsGi. Thus, there arerindicesisuch thatgi(u)andgi(v)are distinct and within distance` inPn. This implies that fi(gi(u))and fi(gi(v))are also distinct and within distance`inG. By the definition ofGk,r,`(n)this implies thatf(u)andf(v)are adjacent inGk,r,`(n) andfis the desired embedding ofGintoGk,r,`(n).

From this, Theorem 2.14 follows immediately forr= 3.

Proof of Theorem 2.14, caser= 3. Note, that the graphG4,3,8(n)hasm⁴=O(n)vertices andO(nm) = O(n^5/4)edges. By Theorem 7.9G_4,3,8(n)isF4,3,8(n)-universal, and sinceF4,3,8(n)hitsF⁽³⁾(n,2)on 82

7. Constructions of universal hypergraphs

P3, Lemma 7.1 implies that HP3,3(G4,3,8(n))is F⁽³⁾(n,2)-universal, has O(n)vertices andO(n^3/2) edges. This proves the caser= 3.

We believe that the constructions from [9] can also be adapted to work with(k, r, `)-decomposable graphs. For the cases discussed here this would lead to universal graphs onnvertices, where the number of edges is some polylog factor larger.

F

^(r)

(n, 2)-universal hypergraphs of uniformity r ≥ 5

Proof of Theorem 2.14 for oddr≥5. First we define the hypergraphHwhich will then turn out to be F^(r)(n,2)-universal. Lett= (r−3)/2. LetG1,. . . ,Gt+1be vertex-disjoint graphs, whereG1, . . . ,Gtare copies ofC_n⁴(the fourth power of the cycleCn) andGt+1is a copy of the graphG4,3,8(n), introduced in the previous section. Furthermore, we add on top ofGt+1another graphG^∗_t+1containing as edges all pairs of vertices which have a common neighbour inG_t+1. We defineHto be ther-graph on the vertex set∪˙^t+1_i=1V(Gi), and the edges arer-element subsetsfsuch that, withfi:=f∩V(Gi), we have

|f_i| ≤3and eachG_i[f_i]contains a copy ofP_|f_i_|, a path on|f_i|vertices (thus,P₀is the empty graph, P1 =K1andP2 =K2). Additionally, in the case|ft+1| = 2, we allowft+1to be an edge (i.e.P2) in G^∗_t+1instead ofGt+1.

Certainly, HhasO(n)vertices. How many edges does the hypergraph Hcontain? For this we need to choose pathsP_`_i from everyG_i(respectivelyG^∗_t+1) such that`_i∈ {0,1,2,3}andPt+1

i=1`_i=r. BecauseG1, . . . ,Gthave maximum degree8,Gt+1has maximum degreeO(n^1/4), andG^∗_t+1has max-imum degreen^1/2, we compute the number of edges ofHto beO(n^t+1n^2/4) =O(n^r/2), as desired.

Given a hypergraphH and a subset of verticesX ⊆V, we denote throughH(X)the (not neces-sarily uniform) hypergraph on the vertex setX, whose edges are restrictions toX, i.e.E(H(X_i)) = {f∩Xi:f ∈E(H)}.

The rest of the proof hinges on the following auxiliary lemma (whose proof can be found below) and the caser= 3of Theorem 2.14 shown in the previous section.

Lemma 7.10. LetH ∈ F^(r)(n,2)andt= (r−3)/2. Then there exists a partition of the vertex set ofHinto disjoint subsetsX₁, . . . , X_t+1, such thatH(X₁), . . . , H(X_t+1)have maximum vertex degree2and contain hyperedges of cardinality at most 3. Moreover in H(X1), . . . , H(Xt)every component contains at most2 hyperedges of size3.

Let us see how thenH can be embedded into the hypergraphH. Owing to the special structure ofH(X₁), . . . , H(X_t), one can easily find injective mapsg_i:X_i →V(G_i), such that every hyperedge f ∈E(H(Xi))is such thatGi[gi(f)]contains a pathP_|f|. This can be seen by replacingf inH(Xi) through an arbitrary pathP_|f|obtaining thus the graphG⁰_ion the vertex setXi. Then, since in ev-ery component ofH(Xi)there are at most two edges of size3, it is easy to find an injective graph homomorphism fromG⁰_iintoGi.

ForH(Xt+1)we can assume first that it is3-uniform and lies inF⁽³⁾(n,2)by adding somedummy vertices appropriately (but still using the notationH(Xt+1)). TheF4,3,8(n)-universality ofGt+1 = G4,3,8(n)and the fact thatF4,3,8(n)hitsH(Xt+1)onP3yields an injective mapgt+1: Xt+1→V(Gt+1) such thatGt+1[gt+1(f)]containsP3 for everyf ∈ E(H(Xt+1)). Deleting the dummy vertices (but keeping the edges) we see thatg_t+1remains injective andG_t+1[g_t+1(f)]containsP_|f|for everyf ∈

7.3 Proof for∆ = 2

E(H(Xt+1))except possibly for the case, when the center vertex of some P3 was deleted (being a dummy vertex). But in this case we observe that G^∗_t+1[gt+1(f)] induces P2 instead, because both vertices ofg_t+1(f)were incident to the deleted vertex inG_t+1.

It should be clear thatg:V(H)→ V(H)withg|_X_i =g_i, for alli ∈[t+ 1], is injective. It remains to show thatg is a homomorphism into H. Given an edge eofH, by the definition ofH(Xi)and the choices ofgi’s, we see thate∩Xi ∈E(H(Xi))andGi[gi(e∩Xi)]contains a pathP_|e∩X_i_|for all i, except possibly for the case when|gt+1(e∩X_t+1)| = 2. But in this case one must necessarily have gt+1(e∩Xt+1)∈E(G^∗_t+1). These conditions fulfill exactly the requirement forg(e)to be the edge in H. Thus,gembedsHintoH.

Finally, we provide the proof for the auxiliary lemma above, Lemma 7.10.

Proof of Lemma 7.10. LetH ∈ F^(r)(n,2). Again we assume first thatH is linear and2-regular. We consider, as in the caser= 3, the line graphH^∗, which isr-regular now. Hence Lemma 7.4 yields a spanning subgraphH₁^∗, in which every vertex has degree2or3and every component has at most2 vertices of degree3.

IfCis a component ofH₁^∗, then we define the setVCas all verticesvsuch that{v}=e∩ffor some ef ∈E(C)(recall thatH is assumed to be a linear hypergraph). We setX1 =∪VCwhere the union is over all componentsC ofH₁^∗ and then the set{v:{v}=e∩f for someef ∈E(C)} is an edge of H(X1)for every edgef ∈ E(H). Observe, that these edges have cardinality either2or3. Indeed, a vertex of degreej in some componentC is the edge ofH that intersectsj other edges of H in different vertices, which give rise to aj-uniform edge inH(X1). By construction,H(X1)is linear and 2-regular. Crucially, the components ofH(X1)have simple structure, since these areinheritedfrom the componentsC. More precisely, each component ofH(X₁)has at most two3-uniform edges and all other edges have cardinality2.

We denote byH˜1 = H(V(H)\X1)the hypergraph obtained fromH by deleting from its edges all vertices fromX1 (we call this procedure as reducing uniformity). It should be clear that, in this way every edge ofH can be written uniquely as the union of one edge ofH(X₁)and the other from H˜1. SinceH(X1)is not necessarily uniform, the hypergraph H˜1 is now a not necessarily uniform hypergraph as well, but its edges have cardinalities eitherr−3orr−2.

The next step calls for an inductive procedure with a blemish, thatH˜1is not necessarily uniform.

But this can be remedied by adding dummyvertices and edges to H˜1 and obtaining an (r − 2)-uniform linear hypergraph still denoted by H˜1 which is 2-regular³⁸. We keep doing this reduc-tion until we arrive at the hypergraphH˜_twheret = (r−3)/2, thereby generating X₂, . . . , X_t and H˜2(X2). . . ,H˜_t−1(Xt). Finally we getXt+1:=V(H)\ ∪^t_i=1Xiand a3-uniform linear hypergraphH˜t

onX_t+1, which is2-regular.

Before we proceed, let us summarise what we achieved so far. We have found hypergraphsH(X1), H˜₂(X₂), . . . ,H˜_t−1(X_t), so that each of them is linear,2-regular and its edge uniformities are either2or 3and each of its components has simple structure (recall: each component has at most two3-uniform edges and all other edges have cardinality2). Furthermore H˜_t is a 3-uniform linear hypergraph, which is2-regular, and the vertex setsX1, . . . , Xt+1are a partition ofV(H).

38Once we are finished with the decomposition, we will reduce the uniformity by deleting these dummy vertices from edges, but keeping the altered edges.

7. Constructions of universal hypergraphs

We finally obtain the promised familyH(X1), . . . , H(Xt+1). This can be seen as reducing uniformi-ties of the hypergraphsH(X1),H˜2(X2), . . . ,H˜_t−1(Xt)andH˜tby deleting dummy edges and dummy

Im Dokument Spanning structures in random graphs and hypergraphs (Seite 86-93)