D. Kühn et al. / Journal of Combinatorial Theory, Series B 103 (2013) 291–305 293
Theorem 4. There exists an n
0
∈ N such that the following holds. Suppose that H is a 3-uniform hypergraph
on n
n
0
vertices, that n/3 d ∈ N and that
δ
1
(H)>
n − 1
2
−
n −d
2
.
Then H contains a matching of size at least d.
It would be interesting to obtain analogous results (i.e. minimum degree conditions which guar-
antee a matching of size d)forr-uniform hypergraphs and for r-partite hypergraphs. Some bounds
are given in [5]. Further, a 3-partite version of Theorem 1 was recently proved by Lo and Mark-
ström [12].
Treglown and Zhao [18,19] determined the minimum
-degree that ensures a perfect matching
in an r-uniform hypergraph when r
/2 r − 1. (Independently, Czygrinow and Kamat [4] dealt
with the case when r
= 4 and = 2.) Prior to this, Pikhurko [14] gave an asymptotically exact result.
The situation for
-degrees where 1 <<r/2isstillopen.In[6], Hàn, Person and Schacht provided
conditions on
δ
(H) that ensure a perfect matching in the case when <r/2. These bounds were
subsequently lowered by Markström and Ruci
´
nski [13]. Alon, Frankl, Huang, Rödl, Ruci
´
nski and Su-
dakov [2] discovered a connection between the minimum
-degree that forces a perfect matching
in an r-uniform hypergraph and the minimum
-degree that forces a perfect fractional matching.As
a consequence of this result they determined, asymptotically, the minimum
-degree that ensures a
perfect matching in an r-uniform hypergraph for the following values of
(r,): (4, 1), (5, 1), (5, 2),
(6, 2) and (7, 3). See [15] for further results concerning perfect matchings in hypergraphs.
2. Notation
Given a hypergraph H and subsets V
1
, V
2
, V
3
of its vertex set V (H), we say that an 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.DefineH
n,d
(V , W )
tobethehypergraphwithvertexsetV ∪ W consisting of all those edges which have type VVW or
VWW.ThusH
n,d
(V , W ) has a matching of size d,
δ
1
H
n,d
(V , W )
=
n − 1
2
−
n −d − 1
2
and H
n,d
(V , W ) is very close to the extremal hypergraph which shows that the degree 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
3
.
In this case we also call V and W vertex classes of H.(SoH does not have unique vertex 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 ,wewriteN
H
(v) for the neighbourhood of v,i.e.
the set of all those (unordered) tuples of vertices which form an edge together with v.Giventwo
disjoint sets A
, B ⊆ V (H),wedefinethelink graph L
v
(A, B) of v with respect to A, B to be the bipartite
graph whose vertex classes are 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),wedefinethelink graph L
v
(A) of v with respect to A to be the
graph whose vertex set is A and in which a
, a
∈ A are joined if and only if aa
∈ N
H
(v). Also, given
disjoint sets A
, B, C , D, E ⊆ V (H),wewriteL
v
(ABCD) for L
v
(A, B) ∪ L
v
(B, C) ∪ L
v
(C, D).Wedefine
L
v
(ABCDE) similarly. If M is a matching in H and E, F are two edges in M with v /∈ E, F ,wewrite
L
v
(EF) for L
v
(V (E), V (F )).IfE
1
,...,E
5
are matching edges avoiding v,wedefineL
v
(E
1
...E
4
) and
L
v
(E
1
...E
5
) similarly. If e = uw is an edge in the link graph of v,thenwewriteve for the edge
vuw of H. A matching in H of size d is called a d-matching.
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