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?

First recursive function I've written in years.

recurse:=function(G,B,i,g,f)
  local j;
  if Size(g) = (i-1) then
    if Size(g) = Size(B) then
      return f(g);
    else
      for j in Combinations(G,B[i]-1) do
        return recurse(G,B,i+1,Concatenation(g,[j]),f);
      od;
    fi;
  else return;
  fi;
end;


global_array:=[];
fun:=function(G,B,g)
  local gset,glist,difflist,gcopy,x,L,i,j,pos,idealdifflist;
#TODO Implement FindL, code's written, just wrap it up
  L:=2;#FindL(G,B);
  gcopy:=StructuralCopy(g);
  gset:=Flat(gcopy);
  glist:=List(G);
  difflist:=ConstantArray(0,Size(G));
  idealdifflist:=ConstantArray(L,Size(G));
  idealdifflist[Position(glist,One(G))]:=0;
  # if Id is not any one of the diffs;
  if Positions(gset,One(G))<>[] then
    return;
  else
    for i in [1..Size(gcopy)] do
      Add(gcopy[i],One(G));
    od;
#generate difflist
    for i in [1..Size(gcopy)] do
      for j in Tuples(gcopy[i],2) do
        pos:=Position(glist,j[1]*j[2]^-1);
        difflist[pos]:=difflist[pos]+1;
      od;
    od;
  fi;
#verify difflist
  if difflist = idealdifflist then
    Add(global_array,g);
  fi;
end;