A theorem on families of sets

AbstractWe prove the following result and transfinite extensions of it: Let (Mi:i ϵ I) be a family of non-zero subsets of the set S. If the cardinalities |I| = f and |S| = n are finite and f > n(r − 1), then one can find r disjoint subsets Iυ(υ = 1,…,r) of I for which ⋃i∈I1Mi = … = ⋃i∈IrMiThe proof is constructive. We apply a generalization by R. Rado of P. Hall's celebrated theorem on systems of representatives.Another proof of the above result has been found by H. Tverberg (see [3]). Tverberg applies his generalization of Radon's theorem (see [2]). He also shows by an example that the result is in a sense best possible