Furthermore, this bound on t can only be achieved by taking the Sk to be all the [~] -element subsets of S, or, n is odd, by taking all the ([~] +l)-element when subsets. Sperner's theorem, like Schur's work [15] on the solutions of x m + ym = z m van der Waerden's theorem [17] on arithmetic progressions, Ramsey's fundamental result [12] on partitions of the subsets o f a set, and Polya's approach [11 ] to the theory o f enumeration, has been the seed f r o m wh ich a major branch of combinatorial theory has grown during the past 50 years. This branch, often called extremal set theory, has been especially active during the past 10 years. In particular, one of the outstanding open problems, which was first raised by G. -C. Rota nearly 15 years ago and which was responsible for much of this activity, has just been settled within the past year by E. Rodney Canfield of the University o f Georgia. What is even more intriguing is that Canfield showed that the answer that everyone had expected (and was trying to prove) was wrong*. In this note I would like to give a brief sketch of the background of Rota's problem and its resolution. By a chain C in a (finite) partially ordered set P we mean a totally ordered subset of P; the length of C is just the number o f elements in it. We say that P is graded if P has a unique minimal element 0 and for every p e P, all maximal chains from 0 to p have the same length, called the rank of p. We denote the elements of P having rank k by ek, also called the k th level. By an anfichain in P we mean a subset of mutually incomparable elements of P. For example, for any k the set e k forms an antichain.