On the density of sets of vectors

As a corollary of the main resu!t of this paper (Theorem i) we show that if ~ i~ not S-dense, then I~[ ~< 1~ s [. This contains as special cases the results of Sauer [4] and Perles and Shelab [5], and of Karpovski and Milman [3]. Before stating Theorem l, we need one more definition; ~ c 3~ is monotone if f ¢ ~ , g e 3 ~ and f ~ g imply g ¢ ~ . Note ~.hat ~ s is monotone and that a monotone family ~ ~3~ is not S-dense iff ~ ~ 3 ~'~. (If P~(f)-~0 for some I~ S a rd f e ~ , then the monotonicity of ~ implies that ~ is I-dt:nse.)