Set families (possibly parameterized) - Combinatorial Parameters

(see [maass1992lower]): define the addressing function $f(S)$ that maps the characteristic vector of the set $S$ to its value as a binary number, then the Addressing family is defined as $$\mathcal{H}:= { {f(S)+1\}\cup S : S \subset {n+1,\ldots,n+\lfloor \log n \rfloor}}$$ In other words, we have singletons on $[n]$ followed by the index of the singleton encoded in binary. So this class has VC $\lfloor\log n\rfloor$. Any parameter that is $p$-monotonic will be larger than its value for singletons on $[n]$ and than a full cube on $[\log n]$. This class also has $coVC=\Omega(n)$