Matchings in 3-uniform hypergraphs
aa r X i v : . [ m a t h . C O ] N ov MATCHINGS IN -UNIFORM HYPERGRAPHS DANIELA K ¨UHN, DERYK OSTHUS AND ANDREW TREGLOWN
Abstract.
We determine the minimum vertex degree that ensures a perfect match-ing in a 3-uniform hypergraph. More precisely, suppose that H is a sufficiently large3-uniform hypergraph whose order n is divisible by 3. If the minimum vertex de-gree of H is greater than (cid:0) n − (cid:1) − (cid:0) n/ (cid:1) , then H contains a perfect matching. Thisbound is tight and answers a question of H`an, Person and Schacht. More generally,we show that H contains a matching of size d ≤ n/ (cid:0) n − (cid:1) − (cid:0) n − d (cid:1) , which is also best possible. This extends a result ofBollob´as, Daykin and Erd˝os. Introduction A perfect matching in a hypergraph H is a collection of vertex-disjoint edges of H which cover the vertex set V ( H ) of H . A theorem of Tutte [20] gives a charac-terisation of all those graphs which contain a perfect matching. On the other hand,the decision problem whether an r -uniform hypergraph contains a perfect matchingis NP-complete for r ≥
3. (See, for example, [7] for complexity results in the area.)It is natural therefore to seek simple sufficient conditions, such as minimum degreeconditions, that ensure a perfect matching in an r -uniform hypergraph. This hasturned out to be a difficult question: despite considerable attention, the full solu-tion remains elusive. But the partial results obtained so far have already involvedthe development of new techniques and uncovered interesting connections to otherproblems.Given an r -uniform hypergraph H and distinct vertices v , . . . , v ℓ ∈ V ( H ) (where1 ≤ ℓ ≤ r −
1) we define d H ( v , . . . , v ℓ ) to be the number of edges containing eachof v , . . . , v ℓ . The minimum ℓ -degree δ ℓ ( H ) of H is the minimum of d H ( v , . . . , v ℓ )over all ℓ -element sets of vertices in H . Of these parameters the two most naturalto consider are the minimum vertex degree δ ( H ) and the minimum collective degree or minimum codegree δ r − ( H ). R¨odl, Ruci´nski and Szemer´edi [17] determined theminimum codegree that ensures a perfect matching in an r -uniform hypergraph.This improved bounds given in [10, 16]. An r -partite version was proved by Aharoni,Georgakopoulos and Spr¨ussel [1].Much less is known about minimum vertex degree conditions for perfect matchingsin r -uniform hypergraphs H . H`an, Person and Schacht [6] showed that the thresholdin the case when r = 3 is (1 + o (1)) (cid:0) | H | (cid:1) . (Here, | H | denotes the number of verticesin H .) This improved an earlier bound given by Daykin and H¨aggkvist [5]. In thispaper we determine the threshold exactly, which answers a question from [6]. Date : October 11, 2018.
Theorem 1.
There exists an n ∈ N such that the following holds. Suppose that H is a -uniform hypergraph whose order n ≥ n is divisible by . If δ ( H ) > (cid:18) n − (cid:19) − (cid:18) n/ (cid:19) then H has a perfect matching. Independently, Khan [8] has given a proof of Theorem 1 using different arguments.The following example shows that the result is best possible: let H ∗ be the 3-uniformhypergraph whose vertex set is partitioned into two vertex classes V and W of sizes2 n/ n/ − W . Then H ∗ does not have a perfect matchingand δ ( H ) = (cid:0) n − (cid:1) − (cid:0) n/ (cid:1) .The example generalises in the obvious way to r -uniform hypergraphs. This leadsto the following conjecture, which is implicit in several earlier papers (see e.g. [6, 11]).Partial results were proved by H`an, Person and Schacht [6] as well as Markstr¨om andRuci´nski [13]. Conjecture 2.
For each integer r ≥ there exists an integer n = n ( r ) such thatthe following holds. Suppose that H is an r -uniform hypergraph whose order n ≥ n is divisible by r . If δ ( H ) > (cid:18) n − r − (cid:19) − (cid:18) ( r − n/rr − (cid:19) , then H has a perfect matching. Recently, Khan [9] proved Conjecture 2 in the case when r = 4. It is also naturalto ask about the minimum (vertex) degree which guarantees a matching of given size d . Bollob´as, Daykin and Erd˝os [3] solved this problem for the case when d is smallcompared to the order of H . We state the 3-uniform case of their result here. Theabove hypergraph H ∗ with W of size d − Theorem 3 (Bollob´as, Daykin and Erd˝os [3]) . Let d ∈ N . If H is a -uniformhypergraph on n > d + 1) vertices and δ ( H ) > (cid:18) n − (cid:19) − (cid:18) n − d (cid:19) then H contains a matching of size at least d . Here we extend this result to the entire range of d . Note that Theorem 4 generalisesTheorem 1, so it suffices to prove Theorem 4. Theorem 4.
There exists an n ∈ N such that the following holds. Suppose that H is a -uniform hypergraph on n ≥ n vertices, that n/ ≥ d ∈ N and that δ ( H ) > (cid:18) n − (cid:19) − (cid:18) n − d (cid:19) . Then H contains a matching of size at least d . ATCHINGS IN 3-UNIFORM HYPERGRAPHS 3
It would be interesting to obtain analogous results (i.e. minimum degree conditionswhich guarantee a matching of size d ) for r -uniform hypergraphs and for r -partitehypergraphs. Some bounds are given in [5]. Further, a 3-partite version of Theorem 1was recently proved by Lo and Markstr¨om [12].Treglown and Zhao [18, 19] determined the minimum ℓ -degree that ensures a per-fect matching in an r -uniform hypergraph when r/ ≤ ℓ ≤ r −
1. (Independently,Czygrinow and Kamat [4] dealt with the case when r = 4 and ℓ = 2.) Prior tothis, Pikhurko [14] gave an asymptotically exact result. The situation for ℓ -degreeswhere 1 < ℓ < r/ δ ℓ ( H ) that ensure a perfect matching in the case when ℓ < r/
2. These boundswere subsequently lowered by Markstr¨om and Ruci´nski [13]. Alon, Frankl, Huang,R¨odl, Ruci´nski and Sudakov [2] discovered a connection between the minimum ℓ -degree that forces a perfect matching in an r -uniform hypergraph and the minimum ℓ -degree that forces a perfect fractional matching . As a consequence of this result theydetermined, asymptotically, the minimum ℓ -degree that ensures a perfect matchingin an r -uniform hypergraph for the following values of ( r, ℓ ): (4 , , , , , Notation
Given a hypergraph H and subsets V , V , V of its vertex set V ( H ), we say thatan edge v v v is of type V V V if v ∈ V , v ∈ V and v ∈ V .Let d ≤ n/ V, W be a partition of a set of n vertices such that | W | = d .Define H n,d ( V, W ) to be the hypergraph with vertex set V ∪ W consisting of all thoseedges which have type V V W or V W W . Thus H n,d ( V, W ) has a matching of size d , δ ( H n,d ( V, W )) = (cid:18) n − (cid:19) − (cid:18) n − d − (cid:19) and H n,d ( V, W ) is very close to the extremal hypergraph which shows that the de-gree condition in Theorem 4 is best possible. V and W are the vertex classes of H n,d ( V, W ).Given ε >
0, a 3-uniform hypergraph H on n vertices and a partition V, W of V ( H ) with | W | = d , we say that H is ε -close to H n,d ( V, W ) if | E ( H n,d ( V, W )) \ E ( H ) | ≤ εn . In this case we also call V and W vertex classes of H . (So H does not have uniquevertex classes.) We say that H is ε -close to H n,d if there is a partition V, W of V ( H )such that | W | = d and H is ε -close to H n,d ( V, W ).Given a vertex v of a 3-uniform hypergraph H , we write N H ( v ) for the neigh-bourhood of v , i.e. the set of all those (unordered) tuples of vertices which form anedge together with v . Given two disjoint sets A, B ⊆ V ( H ), we define the link graph L v ( A, B ) of v with respect to A, B to be the bipartite graph whose vertex classesare A and B and in which a ∈ A is joined to b ∈ B if and only if ab ∈ N H ( v ).Similarly, given a set A ⊆ V ( H ), we define the link graph L v ( A ) of v with respectto A to be the graph whose vertex set is A and in which a, a ′ ∈ A are joined ifand only if aa ′ ∈ N H ( v ). Also, given disjoint sets A, B, C, D, E ⊆ V ( H ), we write DANIELA K ¨UHN, DERYK OSTHUS AND ANDREW TREGLOWN L v ( ABCD ) for L v ( A, B ) ∪ L v ( B, C ) ∪ L v ( C, D ). We define L v ( ABCDE ) similarly.If M is a matching in H and E, F are two edges in M with v / ∈ E, F , we write L v ( EF ) for L v ( V ( E ) , V ( F )). If E , . . . , E are matching edges avoiding v , we define L v ( E . . . E ) and L v ( E . . . E ) similarly. If e = uw is an edge in the link graph of v , then we write ve for the edge vuw of H . A matching in H of size d is called a d -matching .Given a set M and k ≥
2, we write (cid:0) Mk (cid:1) for the set of all k -element subsets of M .Given sets M and M ′ , we write M M ′ for the set of all pairs mm ′ with m ∈ M and m ′ ∈ M ′ .Given two graphs G and G ′ , we write G ∼ = G ′ if they are isomorphic. A bipartitegraph is called balanced if its vertex classes have equal size. By a directed graph wemean a graph whose edges are directed, but we only allow at most two edges betweenany pair of vertices: at most one edge in each direction. We write vw for the edgedirected from v to w . Given disjoint vertex sets V and W of a directed graph, wewrite e ( V, W ) for the number of all those edges which are directed from some vertexin V to some vertex in W . A directed graph G is an oriented graph if it has at mostone edge between any pair of vertices (i.e. if G has no directed cycle of length 2).We will often write 0 < a ≪ a ≪ a to mean that we can choose the constants a , a , a from right to left. More precisely, there are increasing functions f and g suchthat, given a , whenever we choose some a ≤ f ( a ) and a ≤ g ( a ), all calculationsneeded in our proof are valid. Hierarchies with more constants are defined in theobvious way. 3. Preliminaries and outline of proof
Our approach towards Theorem 4 follows the so-called stability approach : we provean approximate version of the desired result which states that the minimum degreecondition implies that either (i) H contains a d -matching or (ii) H is ‘close’ to theextremal hypergraph. The latter implies that H is ‘close’ to the hypergraph H n,d defined in the previous section. This extremal situation (ii) is then dealt with sep-arately. We do this in Section 4, where we prove Lemma 7. The proof of Lemma 7makes use of Theorem 3.The non-extremal case is proved in Section 5. As mentioned earlier, an approxi-mate version of Theorem 1 was proved in [6]. However, we need to proceed somewhatdifferently as the argument in [6] fails to guarantee the ‘closeness’ of H to the ex-tremal hypergraph in case (ii). (But we do use the same general approach and anumber of ideas from [6].)We begin by considering a matching M of maximum size and suppose that | M | < d .We then carry out a sequence of steps, where in each step we show that we caneither find a larger matching (and thus obtain a contradiction), or show that H issuccessively ‘closer’ to H n,d . Amongst others, the following fact from [6] will be usedto achieve this (see Figure 1 for the definitions of B , B , B ). Fact 5.
Let B be a balanced bipartite graph on vertices. • If e ( B ) ≥ then B contains a perfect matching. • If e ( B ) = 6 then either B contains a perfect matching or B ∼ = B . ATCHINGS IN 3-UNIFORM HYPERGRAPHS 5 • If e ( B ) = 5 then either B contains a perfect matching or B ∼ = B , B . PSfragreplacements B B B Figure 1.
The graphs B with e ( B ) ≥ B the base vertices of B and the edgebetween them the base edge of B .The proof of the non-extremal case consists of four main steps. Step 1:
We prove that for all but a constant number of vertices x ∈ V ( H ) \ V ( M ),almost all pairs EF ∈ (cid:0) M (cid:1) are such that L x ( EF ) ∼ = B . (See Claims 1–6.) Step 2:
We then show that this implies that M must have size ‘close’ to d (seeClaim 7). Step 3:
Using Step 1, we show that there are 10 vertices v , . . . , v ∈ V ( H ) \ V ( M ),such that for almost all pairs EF ∈ (cid:0) M (cid:1) not only does L v ( EF ) = · · · = L v ( EF ) ∼ = B but further, for each such pair EF the same vertex x plays the role of the basevertex in E (and the analogous statement holds for F also). (See Claim 11 for theprecise statement.) Step 4:
The information obtained in Steps 2 and 3 is then used to conclude that H is ‘close’ to H n,d (see Section 5.3).To see how Fact 5 can be used in Step 1, suppose for example that x , x and x are unmatched vertices, that E and F are edges in M and that the link graphs L x i ( EF ) are identical (call this graph B ). The minimum degree condition impliesthat, for almost all unmatched vertices x , we have e ( L x ( EF )) ≥
5. So let us assumethis holds for x , x , x . If B contains a perfect matching, it is easy to see that wecan transform M into a (larger) matching which also covers the x i , a contradiction.If B ∼ = B , B , we need to consider link graphs involving more than 2 edges from M in order to obtain a contradiction. If B = B , we can use this to prove that weare ‘closer’ to H n,d . In particular, note that if H = H n,d , then in the above examplewe have B = B .To find a matching which is larger than M , we will often need several verticeswhose link graphs with respect to some set of matching edges are identical (as inthe above example). We can usually achieve this with a simple application of thepigeonhole principle. But for this to work, we need to be able to assume that thenumber of vertices not covered by M is fairly large. This may not be true if e.g. weare seeking a perfect matching. To overcome this problem, we apply the ‘absorbingmethod’ which was first introduced in [17]. The method (as used in [6]) guaranteesthe existence of a small matching M ∗ which can ‘absorb’ any (very) small set of DANIELA K ¨UHN, DERYK OSTHUS AND ANDREW TREGLOWN leftover vertices V ′ into a matching covering all of V ′ ∪ V ( M ∗ ). (The existence of M ∗ is shown using a probabilistic argument.) So if we are seeking e.g. a perfect matching,it suffices to prove the existence of an almost perfect one outside M ∗ . In particular,we can always assume that the set of vertices not covered by M is reasonably large,as otherwise we are done by the following lemma. Lemma 6 (H`an, Person and Schacht [6]) . Given any γ > there exists an integer n = n ( γ ) such that the following holds. Suppose that H is a -uniform hypergraphon n ≥ n vertices such that δ ( H ) ≥ (1 / γ ) (cid:0) n (cid:1) . Then there is a matching M ∗ in H of size | M ∗ | ≤ γ n/ such that for every set V ′ ⊆ V ( H ) \ V ( M ∗ ) with γ n ≥| V ′ | ∈ Z there is a matching in H covering precisely the vertices in V ( M ∗ ) ∪ V ′ . Extremal case
The aim of this section is to show that hypergraphs which satisfy the degree con-dition in Theorem 4 and are close to H n,d contain a d -matching. Lemma 7.
There exist ε > and n ∈ N such that the following holds. Suppose that H is a -uniform hypergraph on n ≥ n vertices and d ≤ n/ is an integer. If • δ ( H ) > (cid:0) n − (cid:1) − (cid:0) n − d (cid:1) and • H is ε -close to H n,d ,then H contains a d -matching. We will first prove the lemma in the case when H is not only close to H n,d , butwhen for every vertex v most of the edges of H n,d incident to v also lie in H . Moreprecisely, given α > H on the same vertex set V ( H ) as H n,d , we say that a vertex v ∈ V ( H ) is α -bad if | N H n,d ( v ) \ N H ( v ) | > αn . Otherwisewe say that v is α -good . So if v is α -good then all but at most αn of the edgesincident to v in H n,d also lie in H . We will now show that if d ≥ n/
150 then anysuch H contains a d -matching. Lemma 8.
Let < α < − and let n, d ∈ N be such that n/ ≤ d ≤ n/ .Suppose that H is a -uniform hypergraph on the same vertex set as H n,d and everyvertex of H is α -good. Then H contains a d -matching. Proof.
Let V and W denote the vertex classes of H n,d of sizes n − d and d respectively.Consider the largest matching M in H which consists entirely of edges of type V V W .Let V ′ denote the set of vertices in V uncovered by M . Define W ′ similarly. For acontradiction we assume that | M | < d . First note that | M | ≥ n/
4. Indeed, to seethis consider any vertex w ∈ W ′ . Since w is α -good but N H ( w ) ∩ (cid:0) V ′ (cid:1) = ∅ , it followsthat | V ′ | ≤ √ αn . Thus | M | = | V \ V ′ | / ≥ ( n − d − √ αn ) / ≥ n/ v , v ∈ V ′ and w ∈ W ′ where v = v . Given a pair E E of distinctmatching edges from M , we say that E E is good for v v w if there are all possibleedges E in H which take the following form: E has type V V W and contains onevertex from { v , v , w } , one vertex from E and one vertex from E . Note that if E E is good for v v w then H has a 3-matching which consists of edges of type V V W and contains precisely the vertices in E , E and { v , v , w } . So if such a pair E E exists, we obtain a matching in H that is larger than M , yielding a contradiction. ATCHINGS IN 3-UNIFORM HYPERGRAPHS 7
Since | M | ≥ n/ (cid:0) n/ (cid:1) > n /
40 pairs of distinct matching edges E , E ∈ M . Since v , v and w are α -good there are at most 3 αn < n /
40 suchpairs E E that are not good for v v w . So one such pair must be good for v v w , acontradiction. (cid:3) We now use Lemma 8 to prove Lemma 7. Our strategy is to obtain a ‘small’matching M in H that covers all ‘bad’ vertices in H . We will construct M in stagesso as to ensure that H − V ( M ) satisfies the hypothesis of Lemma 8. Thus we obtaina ( d − | M | )-matching M ′ of H − V ( M ), and hence a d -matching M ∪ M ′ of H . Proof of Lemma 7.
Let 0 < /n ≪ ε ≪ ε ′ ≪ ε ′′ ≪ ε ′′′ ≪
1. By Theorem 3 wemay assume that d ≥ n/ H is as in the statement of the lemma andlet V and W denote the vertex classes of H of sizes n − d and d respectively. Since H is ε -close to H n,d , all but at most 3 √ εn vertices in H are √ ε -good. Let V bad denotethe set of √ ε -bad vertices in V . Define W bad similarly. So | V bad | , | W bad | ≤ √ εn .Define c := | W bad | , V := V ∪ W bad and W := W \ W bad . Thus a := | V | = n − d + c and b := | W | = d − c . Moreover, δ ( H [ V ]) ≥ δ ( H ) − (cid:18) b (cid:19) − ( a − b > (cid:18) n − (cid:19) − (cid:18) n − d (cid:19) − (cid:18) b (cid:19) − ( a − b. But (cid:0) n − (cid:1) = (cid:0) a − (cid:1) + ( a − b + (cid:0) b (cid:1) and so δ ( H [ V ]) > (cid:18) a − (cid:19) − (cid:18) n − d (cid:19) = (cid:18) a − (cid:19) − (cid:18) a − c (cid:19) . Since c ≤ √ εn we can apply Theorem 3 to obtain a matching M of size c in H [ V ].Let H := H − V ( M ) and V := V \ V ( M ). (Note that if W bad = ∅ then H = H .) So H has vertex classes V and W where | V | = a − c . Since H is ε -closeto H n,d ( V, W ) and 3 c ≤ √ εn ≪ ε ′ n we have that H is ε ′ -close to H | H | ,b ( V , W ).By definition of W all vertices in W are ε ′ -good in H . Furthermore, if a vertex v ∈ V ( H ) is ε ′ -bad in H then v ∈ V and v ∈ V bad ∪ W bad . Let V bad denote theset of such vertices. So | V bad | ≤ √ εn . If V bad = ∅ then we can apply Lemma 8 toobtain a b -matching M in H . We thus obtain a matching M ∪ M of size b + c = d in H . So we may assume that V bad = ∅ .We say that a vertex v ∈ V bad is useful if there are at least ε ′ n pairs of vertices v ′ w ∈ V W such that vv ′ w is an edge in H . Clearly we can greedily select a matching M in H such that m := | M | ≤ | V bad | where M covers all useful vertices andconsists entirely of edges of type V V W . Let H := H − V ( M ), V := V \ V ( M )and W := W \ V ( M ). Then | V | = | V | − m = a − c − m and | W | = b − m .Note that δ ( H ) > (cid:18) n − (cid:19) − (cid:18) n − d (cid:19) ≥ (1 − ε ) − (cid:18) − dn (cid:19) ! n
2= (1 − ε ) (cid:18) dn − d n (cid:19) n − ε ) d (cid:18) n − d (cid:19) . (1) DANIELA K ¨UHN, DERYK OSTHUS AND ANDREW TREGLOWN
Consider any vertex v ∈ V bad \ V ( M ). Since v is not useful, it must lie in more than δ ( H ) − n | V ( H ) \ V ( H ) | − ε ′ n − (cid:18) | W | (cid:19) ( ) ≥ (1 − ε ) d (cid:18) n − d (cid:19) − ε ′ n − ε ′ n − d ≥ d ( n − d ) − εdn − ε ′ n ≥ dn − ε ′ n ≥ ε ′ n edges of H [ V ]. Since | V bad | ≤ √ εn we can greedily select a matching M in H [ V ]of size m := | M | ≤ | V bad | which covers all the vertices in H which lie in V bad .Let H := H − V ( M ) and V := V \ V ( M ). So H has vertex classes V and W where | V | = | V |− m = a − c − m − m . Recall that every vertex in V ( H ) \ V bad is ε ′ -good in H . Since V bad ⊆ V ( M ∪ M ) and | H | − | H | = 3( | M | + | M | ) ≪ ε ′ n ,it follows that every vertex of H is ε ′′ -good. So certainly for every vertex w ∈ W there are at least | V || W | / vw ′ ∈ V W such that vww ′ is an edge in H . Thuswe can greedily find a matching M of size m such that each edge in M has type V W W .Let H := H − V ( M ), V := V \ V ( M ) and W := W \ V ( M ). So H has vertexclasses V and W of sizes | V | = | V |− m = a − c − m − m = n − d − c − m − m and | W | = | W | − m = b − m − m = d − c − m − m . Moreover, every vertex of H is ε ′′′ -good. Thus we can apply Lemma 8 to H to obtain a | W | -matching M in H . But then M ∪ M ∪ M ∪ M ∪ M is a matching of size c + m + m + m + | W | = d in H , as desired. (cid:3) We remark that the only point in the proof of Theorem 4 where we need the fullstrength of the minimum degree condition is when we apply Theorem 3 to find thematching M in the proof of Lemma 7.5. Proof of Theorem 4
Preliminaries.
We first define constants satisfying0 < /n ≪ /C ≪ γ ′′ ≪ γ ′ ≪ γ ≪ ε ′ ≪ ε ≪ η ′ ≪ η ≪ α ′ ≪ α ≪ ρ ′ ≪ ρ ≪ τ ≪ . (2)Let H be a 3-uniform hypergraph on n ≥ n vertices such that δ ( H ) > (cid:18) n − (cid:19) − (cid:18) n − d (cid:19) ≥ (1 − γ ′ ) d ( n − d/ , (3)where d is an integer such that 1 ≤ d ≤ n/
3. (Note that the second inequality in(3) follows from the same argument as (1).) We wish to find a d -matching in H .Note that Theorem 3 covers the case when d ≤ n/ n/ ≤ d ≤ n/ d ≥ n/ − τ n . Since τ ≪
1, (3) gives us that δ ( H ) ≥ (1 / γ ′′ ) (cid:0) n (cid:1) . Soby Lemma 6 there is a matching M ∗ in H of size | M ∗ | ≤ ( γ ′′ ) n/ V ′ ⊆ V ( H ) \ V ( M ∗ ) with ( γ ′′ ) n ≥ | V ′ | ∈ Z there is a matching in H coveringprecisely the vertices in V ( M ∗ ) ∪ V ′ . If n/ ≤ d < n/ − τ n we set M ∗ := ∅ . ATCHINGS IN 3-UNIFORM HYPERGRAPHS 9
In both cases we define H ′ := H − V ( M ∗ ). (So H ′ = H if n/ ≤ d < n/ − τ n .)Thus δ ( H ′ ) ≥ δ ( H ) − γ ′ n . (4)Let M be the largest matching in H ′ . Clearly we may assume that | M | < d . Theo-rem 3 implies that n/ ≤ | M | < d. (5)Let V M := V ( M ) and V := V ( H ′ ) \ V M . So | V | ≤ n − | V M | . If n/ ≤ d < n/ − τ n then | V | > n − d > τ n . Suppose d ≥ n/ − τ n . If | V | ≤ ( γ ′′ ) n , then by definitionof M ∗ , there is a matching M ′ in H containing all but at most two vertices from V ( M ∗ ) ∪ V . But then M ∪ M ′ is a matching in H of size ⌊ n/ ⌋ ≥ d , as desired. Soin both cases we may assume that( γ ′′ ) n ≤ | V | ≤ n − | V M | . (6)5.2. Finding structure in the link graphs.
In this section we show that ‘most’of our link graphs L v ( EF ) with v ∈ V and EF ∈ (cid:0) M (cid:1) are copies of B (recall that B was defined after Fact 5). Claim 1.
There does not exist v v v ∈ (cid:0) V (cid:1) and EF ∈ (cid:0) M (cid:1) such that • L v ( EF ) = L v ( EF ) = L v ( EF ) and • L v ( EF ) contains a perfect matching. Proof.
The proof is identical to the proof of Fact 17 in [6]. We include it here forcompleteness. Let E = { x , x , x } and F = { y , y , y } and suppose x y , x y and x y is a perfect matching in L v ( EF ). Since these edges lie in L v i ( EF ) for each1 ≤ i ≤ v x y , v x y and v x y lie in H ′ . Replacing E and F in M with these edges we obtain a larger matching in H ′ , a contradiction. (cid:3) We will now use Claim 1 to show that only a constant number of vertices v ∈ V have ‘many’ link graphs L v ( EF ) containing perfect matchings. Claim 2.
Let V ′ denote the set of all those vertices v ∈ V for which there are at least εn pairs EF ∈ (cid:0) M (cid:1) such that L v ( EF ) contains a perfect matching. Then | V ′ | ≤ C . Proof.
Let G be the bipartite graph with vertex classes V ′ and (cid:0) M (cid:1) where { v, EF } is an edge in G precisely when L v ( EF ) contains a perfect matching. So G contains atleast | V ′ | εn edges. If | V ′ | ≥ C then there is a pair EF ∈ (cid:0) M (cid:1) such that d G ( EF ) ≥ Cε ≥ · (since 1 /C ≪ ε ). Since there are 2 labelled bipartite graphs with vertexclasses E and F , there are 3 vertices v , v , v ∈ V ′ such that L v ( EF ) = L v ( EF ) = L v ( EF ) and L v ( EF ) contains a perfect matching. This contradicts Claim 1, asrequired. (cid:3) Claim 3.
Let V ′′ denote the set of all those vertices v ∈ V for which there are atleast εn pairs EF ∈ (cid:0) M (cid:1) such that L v ( EF ) ∼ = B , B . Then | V ′′ | ≤ C . Proof.
Suppose for a contradiction that | V ′′ | > C . Given any v ∈ V ′′ , define anauxiliary oriented graph G v as follows: The vertex set of G v is M and given EF ∈ (cid:0) M (cid:1) there is an edge directed from E to F precisely when L v ( EF ) ∼ = B , B where E is the vertex class that contains the isolated vertex in L v ( EF ). Since v ∈ V ′′ , wehave that e ( G v ) ≥ εn .We call a path E . . . E of length 4 in G v suitable if its (directed) edges are E E , E E , E E and E E . Our first aim is to find at least ε ′ n suitable paths in G v . Choose a partition V , V of V ( G v ) such that e G v ( V , V ) ≥ e ( G v ) / ≥ εn / V to V in a random partition of V ( G v ).) Let G ′ v denote the undirected bipartitegraph with vertex classes V and V whose edges are all those edges in G v that areoriented from V to V . Since e ( G ′ v ) ≥ εn / G ′ v contains a subgraph G ′′ v with δ ( G ′′ v ) ≥ d ( G ′ v ) / ≥ εn/
5. Thus we can greedily find at least12 · εn (cid:16) εn − (cid:17) . . . (cid:16) εn − (cid:17) ≥ ε ′ n paths of length 4 in G ′′ v whose endpoints both lie in V . By definition of G ′′ v , each ofthese paths corresponds to a suitable path in G v .Consider a suitable path E . . . E in G v . So L v ( E E ) , L v ( E E ) ∼ = B , B with the isolated vertex in both graphs lying in E . Choose edges e of L v ( E E ) and e of L v ( E E ) such that e and e are disjoint. Since L v ( E E ) ∼ = B , B and E contains the isolated vertex in this graph, there is a 2-matching { e , e } in L v ( E E )that is disjoint from e . Similarly since L v ( E E ) ∼ = B , B and E containsthe isolated vertex in this graph, there is a 2-matching { e , e } in L v ( E E ) that isdisjoint from e . Hence L v ( E E E E E ) contains a 6-matching { e , e , e , e , e , e } .Let G be the bipartite graph with vertex classes V ′′ and the set ( M ) of all ordered5-tuples of elements of M where { v, E E E E E } is an edge in G precisely when E . . . E is a suitable path in G v . So G contains at least | V ′′ | ε ′ n edges.Since | V ′′ | > C there exists E E E E E ∈ ( M ) such that d G ( E E E E E ) ≥ Cε ′ ≥ · . Further, there are at most 2 distinct graphs in the collection of all thosegraphs L v ( E E E E E ) for which v ∈ N G ( E E E E E ). Thus there are 6 vertices v , . . . , v ∈ V ′′ such that v , . . . , v ∈ N G ( E E E E E ) and L v ( E E E E E ) = · · · = L v ( E E E E E ). Let { x y , . . . , x y } be a 6-matching in L v ( E E E E E ).So { v x y , . . . , v x y } is a 6-matching in H ′ . Replacing the edges E , . . . , E in M with { v x y , . . . , v x y } we obtain a larger matching, a contradiction. (cid:3) Claim 4.
Let V ′′′ denote the set of all those vertices v ∈ V which fail to satisfy (7) e ( L v ( V , V M )) ≤ (1 + p γ ′ ) | V || M | . Then | V ′′′ | ≤ C . Proof.
Suppose for a contradiction that | V ′′′ | > C ≥ /γ ′ . Given an edge E in M ,we say that E is good for v ∈ V ′′′ if at least two vertices in E have degree at least 3 ATCHINGS IN 3-UNIFORM HYPERGRAPHS 11 in L v ( E, V ). For every v ∈ V ′′′ , there are at least γ ′ | M | edges in M which are goodfor v . (To see this, suppose there are fewer edges which are good for v . Then e ( L v ( V , V M )) < (1 − γ ′ ) | M | (4 + | V | ) + γ ′ | M | · | V |≤ | M || V | (cid:0) (1 − γ ′ )(1 + γ ′ ) + 3 γ ′ (cid:1) ≤ (1 + p γ ′ ) | V || M | , a contradiction to the fact that v ∈ V ′′′ .) This in turn implies that there are v , v ∈ V ′′′ and an edge E in M which is good for both v and v . Then the definition of‘good’ implies that are disjoint edges e ∈ L v ( E, V ) and e ∈ L v ( E, V ) which donot contain v or v . Now we can enlarge M by removing E and adding v e and v e . This contradiction to the maximality of M proves the claim. (cid:3) Claim 5.
Every vertex v ∈ V \ V ′′′ satisfies e ( L v ( V M )) ≥ (5 − γ ) (cid:18) | M | (cid:19) . Proof.
Suppose v ∈ V \ V ′′′ . Then as e ( L v ( V )) = 0 e ( L v ( V M )) ( ) ≥ δ ( H ) − e ( L v ( V , V M )) − γ ′ n ) , ( ) ≥ (1 − γ ′ ) d ( n − d/ − (cid:16) p γ ′ (cid:17) | V || M | − γ ′ n . Now note that the function d ( n − d/
2) is increasing in d for d ≤ n/
3. So e ( L v ( V M )) ≥ (1 − γ ′ ) | M | (cid:18) n − | M | (cid:19) − (cid:16) p γ ′ (cid:17) ( n − | M | ) | M | − γ ′ n ≥ (cid:18) n | M | − | M | − γ ′ n | M | (cid:19) − (cid:16) n | M | − | M | + p γ ′ n | M | (cid:17) − γ ′ n ) ≥ | M | − p γ ′ | M | ≥ (5 − γ ) (cid:18) | M | (cid:19) , which completes the proof of the claim. (cid:3) Claim 6.
Let V ′′′′ denote the set of all those vertices v ∈ V \ V ′′′ for which thereare at least ηn pairs EF ∈ (cid:0) M (cid:1) such that L v ( EF ) contains at most edges. Then | V ′′′′ | ≤ C . Proof.
Suppose for a contradiction that | V ′′′′ | > C . Let v ∈ V ′′′′ . At most 3 | M | edges vv v in H containing v are such that v and v lie in the same edge E ∈ M .Thus Claim 5 implies that X EF ∈ ( M ) e ( L v ( EF )) ≥ (5 − γ ) (cid:18) | M | (cid:19) − | M | ≥ (cid:18) | M | (cid:19) − γn . (8)Let c denote the number of pairs EF ∈ (cid:0) M (cid:1) such that L v ( EF ) contains at most 4edges. Then c ≥ ηn and so (8) implies that there are at least η ′ n pairs EF ∈ (cid:0) M (cid:1) such that L v ( EF ) contains at least 6 edges. Indeed, suppose that this is not the case.Then X EF ∈ ( M ) e ( L v ( EF )) ≤ c + 9 η ′ n + 5 (cid:20)(cid:18) | M | (cid:19) − c (cid:21) = 5 (cid:18) | M | (cid:19) − c + 9 η ′ n < (cid:18) | M | (cid:19) − γn since γ ≪ η ′ ≪ η . This contradicts (8), as desired.Recall from Fact 5 that a balanced bipartite graph B on 6 vertices that containsat least 6 edges either has a perfect matching or B ∼ = B . Thus, given any v ∈ V ′′′′ there are at least r ≥ η ′ n / ≥ εn pairs E F , . . . , E r F r ∈ (cid:0) M (cid:1) such that either • L v ( E i F i ) contains a perfect matching for all 1 ≤ i ≤ r or, • L v ( E i F i ) ∼ = B for all 1 ≤ i ≤ r .So since | V ′′′′ | > C one of the following holds:( α ) There are more than C vertices v ∈ V ′′′′ for which there are at least εn pairs EF ∈ (cid:0) M (cid:1) such that L v ( EF ) contains a perfect matching.( α ) There are more than C vertices v ∈ V ′′′′ for which there are at least εn pairs EF ∈ (cid:0) M (cid:1) such that L v ( EF ) ∼ = B .In either case we get a contradiction: ( α ) contradicts Claim 2 and ( α ) contradictsClaim 3. (cid:3) Recall from Fact 5 that if B is a balanced bipartite graph on 6 vertices with e ( B ) = 5 then either B contains a perfect matching or B ∼ = B , B . If e ( B ) ≥ B contains a perfect matching or B ∼ = B . Thus Claims 2, 3, 4 and 6together imply that all vertices v ∈ V \ ( V ′ ∪ V ′′ ∪ V ′′′ ∪ V ′′′′ ) satisfy( β ) L v ( EF ) ∼ = B for at least (cid:0) | M | (cid:1) − εn − ηn ≥ (1 − α ′ ) (cid:0) | M | (cid:1) pairs EF ∈ (cid:0) M (cid:1) .Let V ∗ := V \ ( V ′ ∪ V ′′ ∪ V ′′′ ∪ V ′′′′ ). Thus | V \ V ∗ | ≤ C. Moreover, each v ∈ V ∗ satisfies e ( L v ( V M )) ≤ − α ′ ) (cid:18) | M | (cid:19) + 9 α ′ (cid:18) | M | (cid:19) + 3 | M | ≤ α ′ ) (cid:18) | M | (cid:19) . (9)Here the term 3 | M | accounts for the edges which have both endpoints in the samematching edge of M .We can now show that M has almost the required size. (This corresponds to Step 2in the proof outline.) This will be used in Section 5.3 to prove that H is close to H n,d . Claim 7. | M | > d − αn . Proof.
Assume for a contradiction that | M | ≤ d − αn . Consider any v ∈ V ∗ . Then(10) d H ′ ( v ) ( ) , ( ) ≥ (1 − γ ′ ) d ( n − d/ − γ ′ n ≥ d ( n − d/ − γ ′ n . ATCHINGS IN 3-UNIFORM HYPERGRAPHS 13
Also e ( L v ( V )) = 0 since M is maximal. Thus d H ′ ( v ) = e ( L v ( V M )) + e ( L v ( V , V M )) ( ) , ( ) ≤ α ′ ) (cid:18) | M | (cid:19) + (1 + p γ ′ ) | V || M |≤ α ′ ) (cid:18) | M | (cid:19) + (cid:16) | M | ( n − | M | ) + p γ ′ n (cid:17) ≤ | M | ( n − | M | /
2) + √ α ′ n < ( d − αn )( n − d/ αn/
2) + √ α ′ n < d ( n − d/ − γ ′ n , a contradiction to (10), as desired. (In the third line we again used that the function d ( n − d/
2) is increasing in d for d ≤ n/ (cid:3) In the next sequence of claims, we will show that there are vertices v , . . . , v ∈ V ∗ whose link graphs L v i ( V M ) are very similar to each other (see Claim 11 for the precisestatement). (This corresponds to Step 3 in the proof outline.) Claim 8.
Suppose v , . . . , v ∈ V ∗ are distinct vertices such that for some EF ∈ (cid:0) M (cid:1) , L v ( EF ) , . . . , L v ( EF ) ∼ = B . Then L v ( EF ) = · · · = L v ( EF ) . Proof.
We suppose for a contradiction that the claim does not hold. Since there are9 labelled bipartite graphs with vertex classes E and F which are isomorphic to B ,two of the L v i ( EF ) must be the same. So we may assume that L v ( EF ) = L v ( EF )but L v ( EF ) = L v ( EF ). Let E = { x , x , x } and F = { y , y , y } . Suppose E ( L v ( EF )) = E ( L v ( EF )) = { x y , x y , x y , x y , x y } . (So x y is the baseedge of L v ( EF ) and L v ( EF ) as defined after Fact 5.) Since L v ( EF ) = L v ( EF )there is an edge e ∈ L v ( EF ) \ L v ( EF ). We may assume e = x y . Replacing E and F with v x y , v x y and v x y in M we obtain a larger matching, a contradiction. (cid:3) Choose distinct v , . . . , v ∈ V ∗ which will be fixed throughout the remainder ofthe proof. Claim 9.
There is a set E of at least (1 − α ) | M | matching edges E ∈ M such thatfor each E ∈ E there are at least (1 − α ) | M | edges F ∈ M for which L v ( EF ) = · · · = L v ( EF ) ∼ = B . Proof.
By ( β ) and Claim 8 there are at least (1 − α ′ ) (cid:0) | M | (cid:1) pairs EF ∈ (cid:0) M (cid:1) suchthat L v ( EF ) = · · · = L v ( EF ) ∼ = B . This in turn immediately implies the claim. (cid:3) Claim 10.
For every E ∈ E there is a set F E of at least (1 − α ) | M | edges in M such that ( δ ) L v ( EF ) = · · · = L v ( EF ) ∼ = B for each F ∈ F E and ( δ ) in each of the L v ( EF ) with F ∈ F E the same vertex x plays the role of thebase vertex in E . (Recall that the base vertices of B are the vertices ofdegree .) Proof.
Since E ∈ E there is a set F ′ E of at least (1 − α ) | M | edges in M such that L v ( EF ) = · · · = L v ( EF ) ∼ = B for each F ∈ F ′ E . Let F E := F ′ E ∩ E . Then |F E | ≥ (1 − α ) | M | and for each F ∈ F E there are at least (1 − α ) | M | edges F ′ ∈ M for which L v ( F F ′ ) = · · · = L v ( F F ′ ) ∼ = B .We claim that F E satisfies the claim. Certainly F E satisfies ( δ ). Suppose fora contradiction that there are F , F ∈ F E such that the vertex x ∈ E that playsthe role of a base vertex in L v ( EF ) is different from the vertex x ∈ E that playsthe role of a base vertex in L v ( EF ). Let F ′ ∈ M be such that L v ( F F ′ ) = · · · = L v ( F F ′ ) ∼ = B , and F ′ = E, F .Since L v ( EF ) ∼ = B and x = x , there exists a 2-matching { e , e } in L v ( EF )that is disjoint from x . Similarly since L v ( F F ′ ) ∼ = B there exists a 2-matching { e , e } in L v ( F F ′ ). Since x ∈ E is a base vertex in L v ( EF ), there is an edge e from x to the vertex in F that is uncovered by { e , e } . So { e , e , e , e , e } is a5-matching in L v ( F EF F ′ ). We have chosen F , F and F ′ so that L v ( F EF F ′ ) = L v ( F EF F ′ ) = · · · = L v ( F EF F ′ ). Thus M ′ := { v e , v e , v e , v e , v e } is a5-matching in H ′ that contains only vertices from E ∪ F ′ ∪ F ∪ F ∪ { v , v , v , v , v } .Replacing E, F ′ , F and F in M with the edges in M ′ yields a larger matching, acontradiction. (cid:3) Given E ∈ E , we call the unique vertex x ∈ V ( E ) satisfying ( δ ) a bottom vertex .If y ∈ E is such that y = x then we say that y is a top vertex . So each E ∈ E containsone bottom vertex and two top vertices whereas none of the at most α | M | edges in M \ E contains a top or bottom vertex. Claim 11.
There are at least (1 − α ) | M | / pairs EF ∈ (cid:0) M (cid:1) such that ( ε ) L v ( EF ) = · · · = L v ( EF ) ∼ = B ; ( ε ) both E and F contain a bottom vertex w and z respectively; ( ε ) wz is the base edge of L v ( EF ) . Proof.
Consider the directed graph G whose vertex set is M and in which there isa directed edge from E to F if E ∈ E and F ∈ F E . Claims 9 and 10 together implythat G has at least (1 − α ) | M | edges and thus at least (1 − α ) | M | / EF ofvertices in G must be joined by a double edge. But each such pair EF satisfies theclaim. (cid:3) Showing that H is √ ρ -close to H n,d . We have now collected all the informa-tion we need for showing that H is close to H n,d ( V, W ), where W will be constructedfrom the set of bottom vertices in M . More precisely, let W ′ denote the set of all thebottom vertices. So Claims 7 and 9 together imply that(11) d − αn ≤ (1 − α ) | M | ≤ |E| = | W ′ | ≤ | M | ≤ d. Let V ′ denote the set of all the top vertices in H . Thus(12) 2 d − αn ≤ − α ) | M | ≤ | V ′ | = 2 | W ′ | ≤ d. Choose a partition
V, W of V ( H ) such that | W | = d , W ′ ⊆ W , V ′ ⊆ V . Note thatsince (11) implies that | W \ W ′ | ≤ αn , all but at most 2 αn vertices of V lie in V . ATCHINGS IN 3-UNIFORM HYPERGRAPHS 15
Our aim is to show that H is √ ρ -close to H n,d ( V, W ). Note that showing this provesTheorem 4 as we can apply Lemma 7 since we chose ρ ≪ Claim 12. H does not contain an edge of type V ′ V V . Proof.
Suppose that the claim is false and let v ′ vv be an edge of H with v ′ ∈ V ′ and v, v ∈ V . Let E ∈ E be the matching edge containing v ′ . Take any F ∈ F E .Take any 2 vertices from v , . . . , v which are not equal to v or v , call them x and y . Since v ′ is a top vertex of E , it follows that L x ( EF ) contains a 2-matching e , e avoiding v ′ . Note that this is also a 2-matching in L y ( EF ). Now we can enlarge M by removing E, F and adding v ′ vv , xe and ye . This contradicts the maximalityof M and proves the claim. (cid:3) Claim 13. • H contains at least (1 − ρ ′ ) | W ′ || V ′ || V | edges of type W ′ V ′ V . • H contains at least (1 − ρ ′ ) | V | (cid:0) | W ′ | (cid:1) edges of type W ′ W ′ V . • H contains at most ρ ′ | V | (cid:0) | V ′ | (cid:1) edges of type V ′ V ′ V . Proof.
To see the first part of the claim, consider any v ∈ V ∗ and any pair w ′ , v ′ with w ′ ∈ W ′ and v ′ ∈ V ′ . Both w ′ , v ′ could lie in the same matching edge from M , but there are at most 3 | M | such pairs. Also, w ′ , v ′ could lie in a pair E, F ofmatching edges from M for which either L v ( EF ) = B or which does not satisfy( ε )–( ε ) in Claim 11. But ( β ) and Claim 11 together imply that there are at most √ αn such pairs E, F . So suppose next that w ′ , v ′ lie in a pair E, F satisfying L v ( EF ) ∼ = B and ( ε )–( ε ). Then L v ( EF ) , L v ( EF ) , . . . , L v ( EF ) ∼ = B and so L v ( EF ) = L v ( EF ) = · · · = L v ( EF ) by Claim 8. Conditions ( ε ) and ( ε ) nowimply that w ′ v ′ ∈ E ( L v ( W ′ , V ′ )). So e ( L v ( V ′ , W ′ )) ≥ | V ′ || W ′ | − √ αn ≥ (1 − ρ ′ / | V ′ || W ′ | . Summing over all vertices v ∈ V ∗ and using that | V \ V ∗ | ≤ C implies the first partof the claim. The remaining parts of the claim can be proved similarly. (cid:3) Claim 14. H contains at least | W ′ | (cid:0) | V | (cid:1) − ρn edges of type W ′ V V . Proof.
Consider any v ∈ V . By Claim 12 there are no edges in L v ( V ( H )) with oneendpoint in V ′ and the other in V . By (11) there are at most 3 α | M | n ≤ αn edgesin L v ( V ( H )) with one endpoint in V M \ ( V ′ ∪ W ′ ) and the other in V . Furthermore, L v ( V ) contains no edges. Thus, e ( L v ( W ′ , V )) ≥ δ ( H ′ ) − e ( L v ( V M )) − αn ) , ( ) , ( ) ≥ (1 − γ ′ ) d (cid:18) n − d (cid:19) − γ ′ n − α ′ ) (cid:18) | M | (cid:19) − αn ) ≥ (1 − γ ′ ) | M | (cid:18) n − | M | (cid:19) − (5 + √ α ) | M | ≥ | M | ( n − | M | ) − √ α | M | n ≥ | W ′ || V | − ρ ′ n . As earlier, here we use the fact that the function d ( n − d/
2) is increasing in d for d ≤ n/
3. Summing over all vertices v ∈ V ∗ and using the fact that | V \ V ∗ | ≤ C now proves the claim. (cid:3) Claim 15. • H contains at least (1 − ρ ) | W ′ | (cid:0) | V ′ | (cid:1) edges of type W ′ V ′ V ′ . • H contains at least (1 − ρ ) | V ′ | (cid:0) | W ′ | (cid:1) edges of type W ′ W ′ V ′ . Proof.
First note that the last part of Claim 13 implies that all but at most 2 √ ρ ′ n vertices x ∈ V ′ lie in at most √ ρ ′ | V ′ || V | edges of type V ′ V ′ V . Call such vertices x useful . Consider any useful x . Then x ∈ E ′ for some E ′ ∈ E ⊆ M . Further,since x is a top vertex in E ′ , certainly there exists an edge F ′ ∈ M such that L v ( E ′ F ′ ) = L v ( E ′ F ′ ) ∼ = B , where x is not a base vertex in L v ( E ′ F ′ ). So L v ( E ′ F ′ ) contains a 2-matching { e , e } which avoids x .Consider any pair EF ∈ (cid:0) M \{ E ′ ,F ′ } (cid:1) satisfying ( ε )–( ε ). We claim that L x ( EF ) ⊆ L v ( EF ). Indeed, if not then there exist disjoint edges e , e and e such that e ∈ E ( L x ( EF )) and e , e ∈ E ( L v ( EF )). Since L v ( E ′ F ′ ) = L v ( E ′ F ′ ) and since EF satisfies ( ε ) we have that v e , v e , xe , v e and v e are edges in H ′ . Replacing E, F, E ′ , F ′ with v e , v e , xe , v e and v e in M yields a larger matching in H ′ , acontradiction. So indeed L x ( EF ) ⊆ L v ( EF ).There are at least (1 − α ) | M | / − | M | ≥ (1 − α ) | M | / EF ∈ (cid:0) M \{ E ′ ,F ′ } (cid:1) satisfying ( ε )–( ε ). We claim that at most ρ | M | / EF are suchthat L x ( EF ) contains fewer than 5 edges. Indeed, suppose not. Since for such EF , L x ( EF ) ⊆ L v ( EF ) ∼ = B , the number of edges of H which contain x and have noendpoint outside V M is at most4 · ρ | M | / · (1 − α − ρ ) | M | / · α | M | / | M | ≤ (5 + 30 α − ρ ) | M | / . Here the third term accounts for edges between pairs not satisfying ( ε )–( ε ) and thefinal term for edges with 2 vertices in the same matching edge from M . Let us nowbound the number of edges containing x which have an endpoint outside V M . Thereare at most | W ′ | ( n − | M | ) ≤ | M | ( n − | M | ) such edges having an endpoint in W ′ and at most √ αn such edges having an endpoint outside V ′ ∪ W ′ ∪ V . Since H has no edge of type V ′ V V by Claim 12, the only other such edges consist of x , onevertex in V ′ and one vertex in V . But since x is useful the number of such edges isat most √ ρ ′ | V ′ || V | . Thus in total there are at most | M | ( n − | M | ) + 2 √ ρ ′ n edgeswhich contain x and have an endpoint outside V M . So the degree of x in H is atmost(5 + 30 α − ρ ) | M | / | M | ( n − | M | ) + 2 p ρ ′ n ≤ | M | ( n − | M | / − ρ n ≤ d ( n − d/ − ρ n ) , ( ) < δ ( H ) , a contradiction. Thus there are at least (1 − α − ρ ) | M | / EF ∈ (cid:0) M \{ E ′ ,F ′ } (cid:1) satisfying ( ε )–( ε ) such that L x ( EF ) = L v ( EF ) ∼ = B . Let P denote the set ofsuch pairs. ATCHINGS IN 3-UNIFORM HYPERGRAPHS 17
Now consider any pair w ′ , v ′ with w ′ ∈ W ′ and v ′ ∈ V ′ \ { x } . Both w ′ , v ′ couldlie in the same matching edge from M , but there are at most 3 | M | such pairs. Also, w ′ , v ′ could lie in a pair E, F of matching edges which does not belong to P . Butthere at most 5 ρ | M | such pairs w ′ , v ′ . So suppose next that w ′ , v ′ lies in a pair E, F belonging to P . Since L x ( EF ) = L v ( EF ) ∼ = B and EF satisfies ( ε ) and ( ε ) itfollows that w ′ v ′ ∈ E ( L x ( EF )). Thus e ( L x ( W ′ , V ′ )) ≥ (1 − ρ ) | W ′ || V ′ | . Summingover all useful vertices x ∈ V ′ proves the first part of the claim. The second partfollows similarly (the only change is that we consider a pair w ′ , w ′ ∈ W ′ in the finalparagraph). (cid:3) Claims 13–15 together with (11) and (12) now show that H contains all but atmost √ ρn edges of type W V V and
W W V and thus H is √ ρ -close to H n,d ( V, W ).Hence H contains a perfect matching by Lemma 7. Remark.
One can also obtain Theorem 4 by proving the result only in the casewhen d = ⌊ n/ ⌋ . Indeed, suppose that H is as in the theorem. Let a := ⌊ ( n − d ) / ⌋ .Obtain a new 3-uniform hypergraph H ′ from H by adding a new vertices to H suchthat each of these vertices forms an edge with all pairs of vertices in H ′ . It is nothard to check that δ ( H ′ ) > (cid:0) | H ′ |− (cid:1) − (cid:0) | H ′ |−⌊| H ′ | / ⌋ (cid:1) and so H ′ has a matching M ′ of size ⌊| H ′ | / ⌋ . One can then show that M ′ contains at least d edges from H , asdesired. (We thank Peter Allen for suggesting this trick.)However, the proof of Theorem 4 is only slightly simpler in the case when d = ⌊ n/ ⌋ (we do not need Claims 12–14 in this case) and to show that the above trick works,one requires some extra calculations. References [1] R. Aharoni, A. Georgakopoulos and P. Spr¨ussel, Perfect matchings in r -partite r -graphs, Euro-pean J. Combin. (2009), 39–42.[2] N. Alon, P. Frankl, H. Huang, V. R¨odl, A. Ruci´nski and B. Sudakov, Large matchings in uniformhypergraphs and the conjectures of Erd˝os and Samuels, J. Combin. Theory Ser. A (2012),1200–1215.[3] B. Bollob´as, D.E. Daykin and P. Erd˝os, Sets of independent edges of a hypergraph,
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J. London Math. Soc (1947), 107–111.Daniela K¨uhn, Deryk Osthus Andrew TreglownSchool of Mathematics School of Mathematical SciencesUniversity of Birmingham Queen Mary, University of LondonEdgbaston Mile End RoadBirmingham LondonB15 2TT E1 4NSUK UK E-mail addresses: { kuehn,osthus }}