 
Summary: Discrete Mathematics 46 (1983) 199202 199
NorthHolland
NOTE
ON THE DENSITY OF SIn'S OF V]I~CTORS*
Noga ALON
School of Mathematical Sciences, TeIAviv University, FelAviv, Israel
Received 16 February 1982
Revised 30 November 1982
Answering a question of Erd6s, Sauer [4] and indepe~dently Pcrles and Shelah [5] found the
maximal cardinality of a collection ~ of subsets of a se~: N of cardinality n such that for ever/
subset M ~ N of cardinality m I{Cf3 M: C ~ 3b'}l< 2". Karl~)vsky and Milman [3] generalised
this result. Here we give a short proof of these results and further extensions.
Let to be the set of nonnegative integers. For fixed positive integers
n, pt ..... p, put N={1,2 ..... n} and define
3~= @(n, Pt ..... p,) = {f: N * to: f(i) < Pi folr all i ~ N}. (1)
For fe 3~ and I c N let P~(f)fl~: I~ to be the restriction of f to I. For ~ c
define P~(~) = {P~(f): f~ ~}. We say that .~ is Idense if pl(~) = p1(~). If S is a
family of subsets of N, then ~ is Sdense if ~? is /dense for some l e S. The
collection 3~s defined below is clearly not Sdense;:
~s = {f ~ ~: (VI ~ S)(:ti E l)[f(i) > 0]}
