suppose that $\mathcal F\subseteq \bigcup_{j=k+1}^{n}X^{(j)}$ and $|X|=n$. we know that $\mathcal F$ is a sperner family and it's also $H_k$. prove that: $\sum_{B\in \mathcal F}\frac{1}{\dbinom{n-1}{|B|-1}}\le 1$ (15 points)
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Tags: combinatorics proposed, combinatorics
suppose that $\mathcal F\subseteq \bigcup_{j=k+1}^{n}X^{(j)}$ and $|X|=n$. we know that $\mathcal F$ is a sperner family and it's also $H_k$. prove that: $\sum_{B\in \mathcal F}\frac{1}{\dbinom{n-1}{|B|-1}}\le 1$ (15 points)