Help a newb out with Design Theory

I have a design $D=(X,B)$ with parameters $t$-$(v,k,\lambda)$.

Let $P^*=(P,\subset)$ be the poset (P is also an abstract simplicial complex)
formed by the following construction:

1. $X \in P$.
2. $B$ is a subset of $P$. (call these facets)
3. If $Y$ is a subset of $b \in B$, then $Y$ is in $P$.

With poset relation subset.

Obviously, all $t$ subsets of $X$ are in this poset, and they have $\lambda$
facets (blocks) that contain them.

Define a new parameter $c$.

Let$ t\leq c\leq k$.
There is a smallest $i$ such that all $i$ subsets of $X$ are in $P$,
but not all $i+1$ subsets are in $P$. Call this smallest integer $c$.

For complete designs, $c$ is $k$.

Does this parameter have a name? does it have a use?

Without loss we can say $t$ is the greatest such $t$ for this design. If $twhat can we say about the number of facets(blocks) that contain these $c$ subsets? nothing?