Parameters measuring the size of set families - Combinatorial Parameters

For a family $\mathcal{H}\subset 2^\mathcal{X}$, we define the effective VC radius as the largest integer $k$ such that all subsets of size $k$ of the effective range of $\mathcal{H}$, i.e. $\bigcup {h\Delta h': h,h'\in \mathcal{H}}$ are shattered by $\mathcal{H}$.