I'm trying to solve the following question:
Let
$\mathcal{F}\subset\mathcal{P}\left(\left\{ 1,\ldots,n\right\} \right)\backslash\left\{ \emptyset\right\}$
a collection of non-empty subsets of $\left\{ 1,\ldots,n\right\} $
such that $ \sum_{F\in\mathcal{F}}2^{-\left|F\right|}\leq\frac{n}{log\left(n\right)}$
prove there exists a set $B\subseteq\left\{ 1,\ldots,n\right\}$ that does not include any subset from $\mathcal{F}$ and $\left|B\right|\geq\left(\frac{1}{2}-o\left(1\right)\right)n$
my thoughts so far: I've been trying to use the Lemma of Kleitman from Alon, Spencer "Probabilistic Methods" , with $\mathcal{B}$ collection of sets that does not include any subset from $\mathcal{F}$ , thus is a downset.
also, I've been trying to understand the meaning behind the $ \sum_{F\in\mathcal{F}}2^{-\left|F\right|}\leq\frac{n}{log\left(n\right)}$ , the expression inside the sum is the probability of picking a set from all the subsets of $F$ .