ISSN 0032-9460, Problems of Information Transmission, 2015, Vol. 51, No. 1, pp. 25–30.
Pleiades Publishing, Inc., 2015.
Original Russian Text
V.M. Blinovsky, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 1, pp. 29–35.
Fractional Matchings in Hypergraphs
V. M. Blinovsky
Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, S˜ao Paulo, Brazil
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow, Russia
Received February 10, 2014; in ﬁnal form, September 15, 2014
Abstract—We ﬁnd an exact formula for the minimum number of edges in a hypergraph which
guarantees a fractional matching of cardinality s in the case where sn is an integer.
Let H =([n],E)beak-uniform hypergraph with vertex set [n] and a set of edges E ⊂
Taking into account a natural bijection between the set of binary n-tuples and the set 2
identify them in what follows.
A fractional matching of a hypergraph of cardinality s ∈ [0, 1] is a set of nonnegative real
,e∈ E} such that
= s and the n-tuple ¯a =(a
coordinates satisfying the inequalities 0 ≤ a
If s ≤ k/n, then the only hypergraph that has no fractional matching of cardinality s is the
hypergraph without edges.
A fractional matching in the case s = 1 is called a perfect fractional matching. This case was
considered in the paper , where the following result was proved.
Theorem 1. The minimum number M +1 of edges in a hypergraph guaranteeing a perfect
fractional matching satisﬁes the equality
M +1= max
n − a
k − i
This theorem was preceded by a conjecture formulated by Ahlswede and Khachatrian in .
In the present paper we ﬁnd a formula for the minimum number of edges in a hypergraph which
has a fractional matching of cardinality s in the case where sn is an integer. As follows from the
above, we may assume that 1 >s>k/n; we also assume that sn is an integer. We prove the
Theorem 2. The maximum number of edges M(s, n, k) in a hypergraph which has no fractional
matching of cardinality s satisﬁes the equality
M(s, n, k)= max
n − c
k − i
Supported by FAPESP, project nos. 2012/13341-8 and 2013/07699-0, and NUMEC/USP, project