A THEOREM ON INDEPENDENCE

Suppose we are given a family of sets l={S(j),j is-an-element-of J}, w here S(j)=and(i=1)k H(i)(j), and suppose each collection of sets H(i)( j1),...,H(i)(j(k+1)) has a lower bound under the partial ordering defi ned by inclusion, then the maximal size of an independent subcollectio n of l is k. For example, for a fixed collection of half-spaces H-1, . .., H(k) in R(d), we define l to be the collection of all sets of the form [GRAPHICS] where x(i), i=1, ..., k are points in R(d). Then the m aximal size of an independent collection of such sets us k. This leads to a proof of the bound of 2d due to Renyi et al. (1951) for the maxi mum size of an independent family of rectangles in R(d) with sides par allel to the coordinate axes, and to a bound of d+1 for the maximum si ze of an independent family of simplices in R(d) with sides parallel t o given hyperplanes H-1,...,H(d+1).