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===Statement===
We state the version of this lemma found in a work of Balogh, Morris, Samotij, and Saxton.<ref>{{cite journal |last1=Balogh |first1= József |last2=Morris |first2=Robert|last3=Samotij|first3=Wojciech |title=Independent sets in hypergraphs |journal=Journal of the American Mathematical Society |date=2015 |volume=28 |issue= 3 |pages=669–709|doi= 10.1090/S0894-0347-2014-00816-X |s2cid= 15244650 |doi-access=free |arxiv=1204.6530 }}</ref>
Let <math>\mathcal{H}</math> be a <math>k</math>-uniform hypergraph and suppose that for every <math>l \in \{1,2,\ldots, k\}</math> and some <math>b, r \in \mathbb{N}</math>, we have that <math>\Delta_l(H) \le \left( \frac{b}{|V(H)|} \right)^{l-1} \frac{|E(H)|}{r}</math>. Then, there is a collection <math>\mathcal{C} \subset \mathcal{P}(V(H))</math> and a function <math>f
* for every <math>I \in \mathcal{I}(H)</math> there exists <math>S \subset I</math> with <math>|S|\le(k-1)b</math> and <math>I\subset f(S)</math>.
* <math>|C| \le |V(H)| - \delta r</math> for every <math>C \in \mathcal{C}</math> and <math>\delta = 2^{-k(k+1)}</math>.
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