Given the set $\mathcal{F}\in\mathcal{P}(\{1,...,m\})$, I need to provide it with probability $\frac{1}{2}$ a weight function $w:\{1,...,m\}\rightarrow\{1,...,n\}$ such that there will be a single minimal $F\in\mathcal{F}$. The problem is that we are not given what $\mathcal{F}$ and $m$ are. The algorithm needs to provide the $w(i)$ every time it's asked for the weight of some $i$, and never give a number that is bigger than some constant polyinomial of $m$.
I'm required for formulate the appropriate lemma that is needed for this algorithm, and provide a proof (at least where differs from the isolation lemma proof), and describe the algorithm and why it works.
I came across this question why studying probabilistic methods and algorithms, specifically in a chapter on the isolation lemma ( obviously :) ). please excuse the the bad translation, the questions isn't originally in English.