Preface

Summary

Sperner theory is a very lively area of combinatorics and combinatorial optimization. Though nobody knows the exact age of it, we may assume that it was born in the end of the 1920s and that at least F.S. Macaulay and E. Sperner belong to the set of parents. Macaulay's contribution is discussed in the chapter on Macaulay posets. Sperner's theorem is the starting point of this book. It answers the following question: Given a family of pairwise unrelated (with respect to inclusion) subsets of a finite set, what is the maximum size of this family? In the 1930s the set of parents was enlarged by P. Erdõs, C. Ko, and R. Rado, who studied intersection conditions, and in the beginning of the 1950s, R.P. Dilworth entered this set by proving his famous min-max theorem on partially ordered sets. We should not forget the grandfather. At the end of the last century R. Dedekind failed to find an explicit formula for the number of monotone Boolean functions, and this difficult problem was later attacked by several authors. It is impossible to mention all the offspring and descendants of Sperner theory. Many of them can be found in the references. The list of names there shows that many well-known mathematicians of this century have contributed to the development of the theory.

Several aspects make Sperner theory so interesting. The problems often may be clearly and simply formulated. For the solutions one indeed needs creativity, and the techniques are frequently surprising. Concerning applications and methods, there are several unexpected relations to other branches of mathematics, such as programming, algebra, probability theory, number theory, and geometry.