I have a set $ M$ , subsets $ L_1,…,L_m$ and natural numbers $ k,l\leq m$ .

The problem is:Are there l unique indices $ 1\leq i_1,…,i_l\leq m$ , such that

$ \hspace{5cm}\left|\bigcap_{j=1}^{l} L_{i_{j}}\right| \leq k$

Now my question is whether this problem is $ NP$ -complete or not. What irritates me is the two constraints $ l $ and $ k$ because the NP-complete problems that were conceptually close to it that I took a look on (set cover, vertex cover) only have one constraint respectively that also appears in this problem.

I then tried to write a polynomial time algorithm that looks at which of the sets $ L_1,…,L_m$ share more than $ k$ elements with other sets but even if all sets would share more than $ k$ elements with other this wouldn’t mean that their intersection has more than $ k$ elements…

This question kind of comes close but in it there is no restriction on the amount of subsets to use and the size of the intersection should be exactly $ k$ , but maybe this could be useful anyways.

Can somebody further enlighten me ?