2 DANIELA KÜHN, DERYK OSTHUS AND ANDREW TREGLOWNTheorem 1. There exists an n 0 ∈ N such that the follow**in**g holds. Suppose that His a 3-**uniform** hypergraph whose order n ≥ n 0 is divisible by 3. If( ) ( ) n−1 2n/3δ 1 (H) > −2 2then H has a perfect match**in**g.Independently, Khan [8] has given a pro**of** **of** Theorem 1 us**in**g different arguments.The follow**in**g example shows that the result is best possible: let H ∗ be the 3-**uniform**hypergraph whose vertex set is partitioned **in**to two vertex classes V and W **of** sizes2n/3+1 and n/3−1 respectively and whose edge set consists precisely **of** all thoseedges with at least one endpo**in**t **in** W. Then H ∗ does not have a perfect match**in**gand δ 1 (H) = ( n−1) (2 − 2n/3)2 .The example generalises **in** the obvious way to r-**uniform** **hypergraphs**. This leadsto the follow**in**g conjecture, which is implicit **in** several earlier papers (see e.g. [6, 11]).Partial results were proved by Hàn, Person and Schacht [6] as well as Markström andRuciński [13].Conjecture 2. For each **in**teger r ≥ 3 there exists an **in**teger n 0 = n 0 (r) such thatthe follow**in**g holds. Suppose that H is an r-**uniform** hypergraph whose order n ≥ n 0is divisible by r. Ifδ 1 (H) >then H has a perfect match**in**g.( ) n−1−r −1( (r −1)n/rr −1Recently, Khan [9] proved Conjecture 2 **in** the case when r = 4. It is also naturalto ask about the m**in**imum (vertex) degree which guarantees a match**in**g **of** given sized. Bollobás, Dayk**in** and Erdős [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−1 shows that the m**in**imum degree bound isbest possible.Theorem 3 (Bollobás, Dayk**in** and Erdős [3]). Let d ∈ N. If H is a 3-**uniform**hypergraph on n > 54(d+1) vertices and( ) ( ) n−1 n−dδ 1 (H) > −2 2then H conta**in**s a match**in**g **of** size at least d.Hereweextendthis resulttotheentire range**of** d. Note that Theorem4generalisesTheorem 1, so it suffices to prove Theorem 4.Theorem 4. There exists an n 0 ∈ N such that the follow**in**g holds. Suppose that His a 3-**uniform** hypergraph on n ≥ n 0 vertices, that n/3 ≥ d ∈ N and that( ) ( ) n−1 n−dδ 1 (H) > − .2 2Then H conta**in**s a match**in**g **of** size at least d.),

MATCHINGS IN 3-UNIFORM HYPERGRAPHS 3Itwouldbe**in**terest**in**g to obta**in** analogous results (i.e. m**in**imumdegreeconditionswhich guarantee a match**in**g **of** size d) for r-**uniform** **hypergraphs** and for r-partite**hypergraphs**. Some boundsare given **in** [5]. Further, a 3-partite version **of** Theorem 1was recently proved by Lo and Markström [12].Treglown and Zhao [18, 19] determ**in**ed the m**in**imum l-degree that ensures a perfectmatch**in**g **in** an r-**uniform** hypergraph when r/2 ≤ l ≤ r − 1. (Independently,Czygr**in**ow 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/2 is still open. In [6], Hàn, Person and Schacht provided conditionson δ l (H) that ensure a perfect match**in**g **in** the case when l < r/2. These boundswere subsequently lowered by Markström and Ruciński [13]. Alon, Frankl, Huang,Rödl, Ruciński and Sudakov [2] discovered a connection between the m**in**imum l-degree that forces a perfect match**in**g **in** an r-**uniform** hypergraph and the m**in**imuml-degreethat forces aperfect fractional match**in**g. As aconsequence**of** this resulttheydeterm**in**ed, asymptotically, the m**in**imum l-degree that ensures a perfect match**in**g**in** an r-**uniform** hypergraph for the follow**in**g values **of** (r,l): (4,1), (5,1), (5,2), (6,2)and (7,3). See [15] for further results concern**in**g perfect match**in**gs **in** **hypergraphs**.2. NotationGiven a hypergraph H and subsets V 1 ,V 2 ,V 3 **of** its vertex set V(H), we say thatan edge v 1 v 2 v 3 is **of** type V 1 V 2 V 3 if v 1 ∈ V 1 , v 2 ∈ V 2 and v 3 ∈ V 3 .Let d ≤ n/3 and let V,W be a partition **of** a set **of** n vertices such that |W| = d.Def**in**e H n,d (V,W) to be the hypergraph with vertex set V ∪W consist**in**g **of** all thoseedges which have type VVW or VWW. Thus H n,d (V,W) has a match**in**g **of** size d,( ) ( ) n−1 n−d−1δ 1 (H n,d (V,W)) = −2 2and H n,d (V,W) is very close to the extremal hypergraph which shows that the degreecondition **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 3 .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 neighbourhood**of** v, i.e. the set **of** all those (unordered) tuples **of** vertices which form anedge together with v. Given two disjo**in**t sets A,B ⊆ V(H), we def**in**e the l**in**k graphL 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 jo**in**ed to b ∈ B if and only if ab ∈ N H (v).Similarly, given a set A ⊆ V(H), we def**in**e the l**in**k 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 jo**in**ed ifand only if aa ′ ∈ N H (v). Also, given disjo**in**t sets A,B,C,D,E ⊆ V(H), we write