I'm writing a paper about a math puzzle and the thing I'm studying ends up equivalent to finding the following parameter of a bipartite graph G with parts X and Y:

The largest $k$ such that any $k$ vertices in $X$ all have a common neighbor.

I'm trying to figure out if this parameter has a name and if the corresponding decision problem is NP-complete (which I strongly suspect it is).

My best lead so far has been this restatement of the parameter: the largest $k$ such that a set family is $k$-wise intersecting. But none of the papers I've looked at seem interested in finding $k$ -- generally it is given and other results follow.

Does anyone know the name of this parameter (if it exists), or if it is NP-complete?