Parameters measuring the size of set families - Combinatorial Parameters

The Path dimension of a family $\mathcal{H}$, denoted by $P(\mathcal{H})$, 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 full path, i.e. there is an ordering $x_1,\ldots,x_d$ of the elements of $S$ and a subset $T\subset S$ such that $$ \forall 0\le k\le d,\, T\Delta {x_1,\ldots,x_k} \in \mathcal{H}_{|S}\,. $$