A theorem on independence

AbstractSuppose we are given a family of sets C = {S(j), j∈J}, where S(j) = ∩ki=1 Hi(j), and suppose each collection of sets Hi(j1),…,Hi(jk+1) has a lower bound under the partial ordering defined by inclusion, then the maximal size of an independent subcollection of C is k. For example, for a fixed collection of half-spaces H1,…,Hk in Rd, we define C to be the collection of all sets of the form ∩i=1k χi+Hi where χi, i=1,…, k are points in Rd. Then the maximal size of an independent collection of such sets us k. This leads to a proof of the bound of 2d due to Rényi et al. (1951) for the maximum size of an independent family of rectangles in Rd with sides parallel to the coordinate axes, and to a bound of d+1 for the maximum size of an independent family of simplices in Rd with sides parallel to given hyperplanes H1,…,Hd+1