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'''Lower bound:''' The only known lower bound is for ordinary graphs, and shows that (for every $n$ and $\epsilon>1/\sqrt{n}$) there is a hypergraph $H$ with $\rank(H)=2$, such that every $(1+\epsilon)$-cut-sparsifier must have $\Omega(n/\epsilon^2)$ edges {{cite|AndoniCKQWZ-16}}. A different, much simpler proof was devised by Carlson, Kolla, Srivastava, and Trevisan {{cite|CarlsonKST-17}}. Thus, there is a large gap between the upper bound $O_\epsilon(n^2)$ and the lower bound $\Omega_\epsilon(n)$ (this notation omits dependence on $\epsilon$).  
 
'''Lower bound:''' The only known lower bound is for ordinary graphs, and shows that (for every $n$ and $\epsilon>1/\sqrt{n}$) there is a hypergraph $H$ with $\rank(H)=2$, such that every $(1+\epsilon)$-cut-sparsifier must have $\Omega(n/\epsilon^2)$ edges {{cite|AndoniCKQWZ-16}}. A different, much simpler proof was devised by Carlson, Kolla, Srivastava, and Trevisan {{cite|CarlsonKST-17}}. Thus, there is a large gap between the upper bound $O_\epsilon(n^2)$ and the lower bound $\Omega_\epsilon(n)$ (this notation omits dependence on $\epsilon$).  
  
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'''Remark:''' The above considers the number of hyperedges to be the size measure. An alternative measure is the total size of all hyperedges, i.e., $\sum_{e\in E} |e|$. The current upper bound on the total size of the sparsifier is $O_\epsilon(n^3)$, because every hyperedge has size at most $n$, and the lower bound is $\Omega_\epsilon(n)$ (coming from ordinary graphs), hence the gap here is even larger.
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'''Remark:''' The above considers the number of hyperedges to be the size measure. An alternative measure is the total size of all hyperedges, i.e., $\sum_{e\in E} |e|$. The current upper bound on the total size of the sparisifier is $O_\epsilon(n^3)$, because every hyperedge has size at most $n$, and the lower bound is $\Omega_\epsilon(n)$ (coming from ordinary graphs), hence the gap here is even larger.

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