Set families (possibly parameterized) - Combinatorial Parameters

(see [maass1992lower]): Define the function $f(i):=\min {\ell: i\le \sum_{j=1}^\ell n2^{-j} + \lceil \log n \rceil + 1}$. The class of tagged singletons is defined as $$ \mathcal{H}:={\emptyset} \cup {{n+\ell\}:\ell \in [\lceil\log n\rceil]} \cup {{i,n+f(i)\}:i\in[n]} $$

So we have the empty set, singletons on $[n+1, n+\lceil\log n\rceil]$, and then singletons on $[n]$ with a tag such that more than half have the first tag, more than one fourth have the second tag and so on.