LYM inequality

Let $ℱ$ be a Sperner family, that is, the collection ofsubsets of $\{1,2,\mathrm{\dots },n\}$ such that no set contains any other subset.Then

$$\sum _{X\in ℱ}\frac{1}{\left(\genfrac{}{}{0pt}{}{n}{\left|X\right|}\right)}\le 1.$$

This inequality is known as LYM inequality by the names ofthree people that independently discovered it:Lubell[2], Yamamoto[4],Meshalkin[3].

Since $\left(\genfrac{}{}{0pt}{}{n}{k}\right)\le \left(\genfrac{}{}{0pt}{}{n}{\lfloor n/2\rfloor }\right)$ for every integer $k$, LYM inequality tells us that $\left|F\right|/\left(\genfrac{}{}{0pt}{}{n}{\lfloor n/2\rfloor }\right)\le 1$ which is Sperner’s theorem (http://planetmath.org/SpernersTheorem).