Parameters measuring the size of set families - Combinatorial Parameters

The Star dimension of a family $\mathcal{H}$, denoted by $S(\mathcal{H})$, 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 $d$-star, i.e. there is a subset $T\subset S$ such that $$ T\in \mathcal{H}_{|S} \,\mbox{ and }\, \forall x\in S,\, T\Delta {x} \in \mathcal{H}_{|S}\,. $$