 
Summary: Nearly perfect matchings in regular simple hypergraphs
Noga Alon
JeongHan Kim
Joel Spencer
Abstract
For every fixed k 3 there exists a constant ck with the following property. Let H be a k
uniform, Dregular hypergraph on N vertices, in which no two edges contain more than one com
mon vertex. If k > 3 then H contains a matching covering all vertices but at most ckND1/(k1)
.
If k = 3, then H contains a matching covering all vertices but at most c3ND1/2
ln3/2
D. This
improves previous estimates and implies, for example, that any Steiner Triple System on N ver
tices contains a matching covering all vertices but at most O(N1/2
ln3/2
N), improving results by
various authors.
1 Introduction
A Hypergraph is a pair (V, H), where V is a finite set of vertices and H is a finite family of subsets
of V , called edges. It is kuniform if every edge contains precisely k vertices. The degree deg(x) is
