 
Summary: Graphs and Combinatorics 1, 1321 (1985)
Graphsand
Combinatorics
9 SpringerVerlag 1985
The Maximum Number of Disjoint Pairs
in a Family of Subsets
N. Alon 1. and P. Frankl 2
1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA
2 U.E.R. Math6matiques, Universit6 de Paris, VII, Paris, 75005 France
Abstract. Let .~"be a family of 2"+1subsets of a 2nelement set. Then the number of disjoint pairs
in ~" is bounded by (1 + o(1))22".This proves an old conjecture of Erd/Ss. Let ~" be a family of
2tl/tk+l~+~" subsets of an nelement set. Then the number of containments in ,~ is bounded by
/ I~1
'X
(1  1/k + o(1))k 2 :" This verifiesa conjecture ofDaykin and Erd/Ss.Asimilar Erd6sStone type
result is proved for the maximum number of disjoint pairs in a family of subsets.
I. Introduction
Let : be a family of m distinct subsets of X = {1, 2..... n}. Let d(,~) (c(:)) denote
the number of disjoint (comparable, respectively) pairs in ~. That is:
