 
Summary: On a hypergraph matching problem
Noga Alon
Raphael Yuster
Abstract
Let H = (V, E) be an runiform hypergraph and let F 2V
. A matching M of H is (, F)
perfect if for each F F, at least F vertices of F are covered by M. Our main result is
a theorem giving sufficient conditions for an runiform hypergraph to have a (1  , F)perfect
matching. As a special case of our theorem we obtain the following result. Let K(n, r) denote
the complete runiform hypergraph with n vertices. Let t and r be fixed positive integers where
t r 2. Then, K(n, r) can be packed with edgedisjoint copies of K(t, r) such that each
vertex is incident with only o(nr1
) unpacked edges. This extends a result of Ršodl [9].
1 Introduction
A hypergraph H is an ordered pair H = (V, E) where V is a finite set (the vertex set) and E is a
family of distinct subsets of V (the edge set). A hypergraph is runiform if all edges have size r.
In this paper we only consider runiform hypergraphs where r 2 is fixed. A subset M E(H) is
a matching if every pair of edges from M has an empty intersection. A matching is called perfect
if M = V /r. A vertex v V is covered by the matching M if some edge from M contains v.
Let F 2V and let 0 1. A matching M is (, F)perfect if for each F F, at least F
