Suppose that each of the elements of S = {1,2,... ,n}is to be colored either red or blue....

Suppose that each of the elements of S = {1,2,... ,n}is to be colored either red or blue. Show that if A1,..., Ar are subsets of S, there is a way of doing the coloring so that at most  of these subsets have all their elements the same color (where |A| denotes the number of elements in the set A).