  Let $G$ be a simple graph with vertex set $V(G)$. A subset $S$ of $V(G)$ is
independent if no two vertices from $S$ are adjacent. The graph $G$ is known to
be a Konig-Egervary (KE in short) graph if $\alpha(G) + \mu(G)= |V(G)|$, where
$\alpha(G)$ denotes the size of a maximum independent set and $\mu(G)$ is the
cardinality of a maximum matching. Let $\Omega(G)$ denote the family of all
maximum independent sets. A collection $F$ of sets is an hke collection if
$|\bigcup \Gamma|+|\bigcap \Gamma|=2\alpha$ holds for every subcollection
$\Gamma$ of $F$. We characterize an hke collection and invoke new
characterizations of a KE graph. We prove the existence and uniqueness of a
graph $G$ such that $\Omega(G)$ is a maximal hke collection. It is a bipartite
graph. As a result, we solve a problem of Jarden, Levit and Mandrescu
\cite{jlm}, proving that $F$ is an hke collection if and only if it is a subset
of $\Omega(G)$ for some graph $G$ and $|\bigcup F|+|\bigcap F|=2\alpha(F)$.
Finally, we show that the maximal cardinality of an hke collection $F$ with
$\alpha(F)=\alpha$ and $|\bigcup F|=n$ is $2^{n-\alpha}$.
