 
Summary: Large matchings in uniform hypergraphs and the conjectures of
Erdos and Samuels
Noga Alon
Peter Frankl
Hao Huang
Vojtech R¨odl §
Andrzej Ruci´nski ¶
Benny Sudakov
Abstract
In this paper we study conditions which guarantee the existence of perfect matchings and per
fect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture
by Erdos on estimating the maximum number of edges in a hypergraph when the (fractional)
matching number is given, which we are able to solve in some special cases using probabilistic
techniques. Based on these results, we obtain some general theorems on the minimum ddegree
ensuring the existence of perfect (fractional) matchings. In particular, we asymptotically deter
mine the minimum vertex degree which guarantees a perfect matching in 4uniform and 5uniform
hypergraphs. We also discuss an application to a problem of finding an optimal data allocation
in a distributed storage system.
1 Introduction
A kuniform hypergraph or a kgraph for short, is a pair H = (V, E), where V := V (H) is a finite
