 
Summary: Discrete Mathematics 58 (1986) 191193
NorthHolland
191
NOTE
EXPLICIT CONSTRUCTION OF EXPONENTIAL SIZED
FAMILIES OF kINDEPENDENT SETS
N. ALON*
Department of Mathematics, MassachusettsInstituteof Technology,
Cambridge, MA 02139, U.S.A.
Received 22 October 1984
Error correcting codes are used to describe explicit collections Fkof subsets of {1, 2,... n},
with IFkl > 2ckn (ck > 0), such that for any selections A, B of kl and k2 of members of Fkwith
kl + k2 = k, there are elements in all the members of A and not in the members of B. This
settles a problem of Kleitman and Spencer and a similar problem of Kleitman, Shearer and
Sturtevant.
1. Introduction
A collection F of subsets of N = {1, 2,..., n} is kindependent if for every k
distinct members A 1, A2, · · ·, Ak of F all 2k intersections Ak=l Bj are nonempty,
where each Bj can be either Aj or its complement Aj. Kleitman and Spencer [4]
proved that for every fixed k there exists a kindependent collection Fk on n
