 
Summary: Families in Which Disjoint Sets Have
Large Union
N. ALON" AND P. FRANKLb
Department of Mathematics
Tel Aviv University
69978 Tel Aviv, Israel
and
Bell Communications Research
Morristown, New Jersey 07960
'AT&T Bell Laboratories
Murray Hill, New Jersey 07971
INTRODUCTION AND STATEMENT OF THE RESULTS
Let X = {1, 2, ...,n} and F be a family of subsets of X,that is F c 2x. For
1I:i I:j In set Ci.17 = {i, ...,j } . For integers k, m with k 2 2, 0 I m I n, we say
that F has property P(k, m)if any k pairwise disjoint members of F have union of
size greater than m.Thus P(k, n)means simply that F contains no k pairwise disjoint
sets.
Let us write m in the form m = kt  r, where 1I r I k. Define
F(n, k, m)= {F E X:IF I +IF n [1, r  11I 2 t).
It is easy to check that F(n, m, k) has property P(k, n). In fact, if F,, ..., Fk are
