Sperner’s theorem

What is the size of the largest family $ℱ$ of subsets ofan $n$-element set such that no $A\in ℱ$ is a subset of$B\in ℱ$? Sperner [3] gave ananswer in the following elegant theorem:

Theorem.

For every family $\mathrm{F}$ of incomparable subsets of an$n$-set, $\mathrm{\left|}\mathrm{F}\mathrm{\right|}\mathrm{\le }\mathrm{\left(}\genfrac{}{}{0pt}{}{n}{\mathrm{\lfloor }n\mathrm{/}\mathrm{2}\mathrm{\rfloor }}\mathrm{\right)}$.

A family satisfying the conditions of Sperner’s theorem is usuallycalled Sperner family or antichain. The laterterminology stems from the fact that subsets of a finite setordered by inclusion form a Boolean lattice.

There are many generalizations of Sperner’s theorem. On one hand,there are refinements like LYM inequality that strengthen thetheorem in various ways. On the other hand, there aregeneralizations to posets other than the Boolean lattice. For acomprehensive exposition of the topic one should consult awell-written monograph by Engel[2].