Parameters measuring the size of set families - Combinatorial Parameters

The coVC number of a family $\mathcal{H}$, denoted by $\covcmod(\mathcal{H})$, is the largest integer $d$ such that there exists a set $S\subset \mathcal{X}$, of size $d$ such that $\mathcal{H}_{|S}$ contains a hollow $d$-star, i.e. there is a subset $T\subset S$ such that $$ T\notin \mathcal{H}_{|S} \,\mbox{ and }\, \forall x\in S,\, T\Delta {x} \in \mathcal{H}_{|S}\,. $$