Parameters measuring the size of set families - Combinatorial Parameters

The Threshold dimension of a family $\mathcal{H}$, denoted by $T(\mathcal{H})$, is the largest integer $d$ such that there exists a set $S\subset \mathcal{X}$, of size $d$ such that $\mathcal{H}_{|S}$ contains thresholds, i.e. there is an ordering $x_1,\ldots,x_d$ of the elements of $S$ such that $$ \forall 0\le k\le d,\, {x_1,\ldots,x_k} \in \mathcal{H}_{|S}\,. $$

Note that this means $\mathcal{H}_{|S}$ contains a $d+1$-chain, i.e. $d+1$ sets $S_1,\ldots,S_{d+1}$ such that $S_1\subset \ldots\subset S_{d+1}$.